Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?
We have $$S_n = \sum_{k=n+1}^{\infty} \dfrac1{k^3} = \int_{n^+}^{\infty} \dfrac{d \lfloor t \rfloor}{t^3} = \left. \dfrac{\lfloor t \rfloor}{t^3} \right \vert_{t=n^+}^{\infty} + 3 \int_{n^+}^{\infty} \dfrac{\lfloor t\rfloor dt}{t^4} = -\dfrac1{n^2}+3 \int_{n^{+}}^{\infty} \dfrac{t-\{t\}}{t^4} dt$$ Hence, we get that $$S_n = -\dfrac1{n^2} + 3 \int_{n^+}^{\infty} \dfrac{dt}{t^3} - 3 \int_{n^+}^{\infty} \dfrac{\{t\}}{t^4}dt = \dfrac1{2n^2} - 3 \underbrace{\int_{n^+}^{\infty} \dfrac{\{t\}}{t^4}dt}_{\mathcal{O}(1/n^3)}$$ You can get better approximations by repeating the above procedure, and this is called as Euler-Maclaurin Summation. To get a higher order approximation, we need to get a good approximation of $\displaystyle \int_{n^+}^{\infty} \dfrac{\{t\}}{t^4}dt$. This is done as follows. $$\int_{n^+}^{\infty} \dfrac{\{t\}}{t^4}dt = \int_{n^+}^{\infty} \dfrac{1/2}{t^4}dt+ \int_{n^+}^{\infty} \dfrac{\{t\}-1/2}{t^4}dt = \dfrac1{6n^3}+\int_{n^+}^{\infty} \dfrac{B_1(\{t\})dt}{t^4}\\ = \dfrac1{6n^3} + \overbrace{\left.\dfrac{B_2(\{t\})}{2t^4} \right \vert_{t=n^+}^{\infty}}^0 + 4 \underbrace{\int_{n^+}^{\infty} \dfrac{B_2(\{t\})}{t^5} dt}_{\mathcal{O}(1/n^4)}$$ where $B_n(x)$ are Bernoulli polynomials of order $n$. Hence, a better asymptotic for $S_n$ is $$S_n = \dfrac1{2n^2} - \dfrac1{2n^3} + \mathcal{O}(1/n^4)$$ Crank this repeatedly to get higher order estimates.
The Euler MacLaurin for infinite sums, where the integrand is well-behaved (by which I mean all the derivatives of the function vanish as $x \to \infty$) is $$\sum_{k=n}^{\infty} f(k) = \int_{n}^{\infty} f(x)dx + \dfrac{f(n)}2 - \sum_{k=1}^{\infty} \dfrac{B_{2k}}{(2k)!} f^{(2k-1)'}(n)$$ where $B_{2k}$ are the Bernoulli numbers.
In your case, $f(x) = 1/x^3$ and $f^{(2k-1)'}(x) = -\dfrac{(2k+1)!}{2 \cdot x^{2k+2}}$.
Hence, we get that $$\sum_{k=n}^{\infty} \dfrac1{k^3} = \dfrac1{2n^2} + \dfrac1{2n^3} + \sum_{k=1}^{\infty} \dfrac{(2k+1)B_{2k}}{2 \cdot n^{2k+2}}$$ $$\sum_{k=n+1}^{\infty} \dfrac1{k^3} = \dfrac1{2n^2} - \dfrac1{2n^3} + \sum_{k=1}^{\infty} \dfrac{(2k+1)B_{2k}}{2 \cdot n^{2k+2}}$$
Not to pile on, but there is a neat expression for finite Riemann zetas, based on the following fact:
$$\int_0^{\infty} dt \frac{t^2 e^{-N t}}{e^t-1} = 2 \sum_{k=N+1}^{\infty} \frac{1}{k^3}$$
You can prove this using a geometric sum-type expansion of the denominator and evaluation of the subsequent integrals. Applying integration by parts, you get the following expansion for large $N$:
$$\int_0^{\infty} dt \frac{t^2 e^{-N t}}{e^t-1} = \frac{1}{N^2} - \frac{1}{N^3} + \frac{1}{2 N^4} + O \left ( \frac{1}{N^5} \right )$$
Therefore a good approximation for the sum is
$$\sum_{k=1}^N \frac{1}{k^3} \approx \zeta(3)-\left[\frac{1}{2 N^2} - \frac{1}{2 N^3} + \frac{1}{4 N^4}\right]$$
Even for small $N$, say, $N=5$, the relative error in the the above approximation is vanishingly small, i.e., less than $0.03\%$. For larger $N \sim 1000$, the error is swamped by machine precision. For even smaller errors, one may use Richardson extrapolation as discussed in Bender & Orszag, p. 375.
The simplest estimate, and a surprisingly accurate one, for some $f(n)$ is $$ \int_{n - 1/2}^{n + 1/2} \; \; f(x) dx. $$
I edited your question to correct, anyway $$ \sum_{k+1}^\infty \frac{1}{n^3} \approx \int_{k + 1/2}^\infty \; \; x^{-3} dx = \frac{2}{(2k+1)^2}, $$ and $$ \sum_{k}^\infty \frac{1}{n^3} \approx \int_{k - 1/2}^\infty \; \; x^{-3} dx = \frac{2}{(2k-1)^2}, $$ this latter value being slightly larger.
If you do some more careful work, you find that the shift by 1/2 gives very attractive small error, because the first derivative function is integrated out over each interval of unit length. So the error is or order $n^{-4},$ with very little effort. Using the Taylor expansion around $x=n,$ I get
$$ \int_{n - 1/2}^{n + 1/2} \; \; f(x) dx \approx f(n) + \frac{f''(n)}{12}. $$ Since $f(x) = x^{-3},$ we find $f''(x) = 12 x^{-5},$ and the sum of $n^{-5}$ out to $\infty$ is approximately a constant times $n^{-4}.$
=-=-=-=-=-=-=-=
const double zeta_3 = 1.20205690315959428539 ;
for (int n = 1; n <= bound; n += 1)
{
sump += 1.0 / (n * n * n);
double diff = zeta_3 - sump;
double est = 2.0 / ( (2.0 * n + 1.0 ) * (2.0 * n + 1.0 ) );
cout.precision(16);
cout << n << " " << sump << " " << diff << " " << est << " " << log(est - diff) / log(1.0 * n) << endl;
} // for n
n sump diff estimate log ratio
1 1 0.2020569031595942 0.2222222222222222 -inf
2 1.125 0.07705690315959424 0.08 -8.408449270798803
3 1.162037037037037 0.04001986612255726 0.04081632653061224 -6.494860116575991
4 1.177662037037037 0.02439486612255726 0.02469135802469136 -5.859859838755892
5 1.185662037037037 0.01639486612255725 0.01652892561983471 -5.540584559448334
6 1.190291666666667 0.01176523649292771 0.01183431952662722 -5.346812200208313
7 1.19320711856171 0.008849784597883881 0.008888888888888889 -5.215697321661032
8 1.19516024356171 0.006896659597883881 0.006920415224913495 -5.12045706241989
9 1.196531985674193 0.005524917485401071 0.00554016620498615 -5.047738467187578
10 1.197531985674193 0.004524917485401181 0.00453514739229025 -4.990128319146853
11 1.198283300475095 0.003773602684499666 0.003780718336483932 -4.9431740534746
12 1.198862004178798 0.003194898980795946 0.0032 -4.904035407248641
13 1.199317170314438 0.002739732845156384 0.002743484224965706 -4.870812283354566
14 1.199681601801318 0.002375301358275905 0.002378121284185493 -4.842183533928056
15 1.199977898097615 0.002079005061979666 0.002081165452653486 -4.817200759352243
16 1.200222038722615 0.001834864436979666 0.001836547291092746 -4.795164613680947
17 1.200425580346877 0.001631322812717606 0.00163265306122449 -4.775547458961447
18 1.200597048110937 0.00145985504865731 0.001460920379839299 -4.757943264762034
19 1.200742841958437 0.001314061201157735 0.001314924391847469 -4.742034155780267
20 1.200867841958436 0.001189061201157804 0.001189767995240928 -4.727567477547702
21 1.200975821658253 0.001081081501341341 0.001081665765278529 -4.714339704360496
22 1.201069736008366 0.0009871671512287072 0.0009876543209876543 -4.70218491428755
23 1.201151925537419 0.0009049776221747852 0.0009053870529651426 -4.690966382055557
24 1.201224263500382 0.0008326396592117646 0.0008329862557267805 -4.680570345424018
25 1.201288263500383 0.0007686396592117006 0.0007689350249903883 -4.670901314294367
26 1.201345159267337 0.0007117438922568109 0.000711997152011392 -4.661878492460928
27 1.201395964530763 0.0006609386288314312 0.0006611570247933885 -4.653433014433951
28 1.201441518466623 0.0006153846929712881 0.0006155740227762388 -4.645505789221971
29 1.20148252055773 0.0005743826018647091 0.0005745475438092502 -4.638045798396245
30 1.201519557594767 0.0005373455648276515 0.0005374899220639613 -4.631008740248393
31 1.201553124779487 0.0005037783801074003 0.0005039052658100278 -4.624355942352548
32 1.201583642357612 0.0004732608019824003 0.0004733727810650888 -4.61805347945706
33 1.201611468831719 0.0004454343278750272 0.0004455335263978615 -4.612071454176108
34 1.201636911534752 0.0004199916248421864 0.0004200798151648813 -4.606383404809224
35 1.201660235149912 0.0003966680096818553 0.0003967466772465781 -4.600965818754339
36 1.20168166862042 0.0003752345391743184 0.0003753049352598987 -4.595797723054888
37 1.201701410787715 0.0003554923718791514 0.0003555555555555556 -4.590860344914228
38 1.201719635018653 0.0003372681409417044 0.000337325012649688 -4.586136825474815
39 1.201736493023676 0.0003204101359179923 0.0003204614645088928 -4.581611975945853
40 1.201752118023676 0.0003047851359179177 0.0003048315805517452 -4.577272072501593
41 1.201766627389472 0.0002902757701221947 0.0002903178980984177 -4.573104680223196
42 1.201780124851949 0.0002767783076451646 0.0002768166089965398 -4.569098502954018
43 1.201792702360848 0.0002642007987465611 0.0002642356982428326 -4.56524325033338
44 1.201804441654612 0.0002524615049823709 0.0002524933720489837 -4.561529528093542
45 1.201815415591512 0.0002414875680825102 0.0002415167250332085 -4.557948742050555
46 1.201825689282644 0.0002312138769506866 0.0002312406058503873 -4.554493008171014
47 1.201835321059803 0.0002215820997915063 0.000221606648199446 -4.551155084738899
48 1.201844363305173 0.0002125398544210455 0.0002125624402168137 -4.547928300412313
49 1.201852863164925 0.000204039994668781 0.0002040608101214162 -4.544806505967626
50 1.201860863164925 0.000196039994668773 0.0001960592098813842 -4.541784013918732
51 1.201868401743602 0.0001885014159923593 0.0001885191818267509 -4.538855561589176
52 1.201875513714471 0.0001813894451230258 0.00018140589569161 -4.53601627043618
53 1.201882230668736 0.0001746724908586739 0.0001746877456546423 -4.533261605315103
54 1.201888581326664 0.0001683218329304736 0.000168335998653312 -4.530587353600314
55 1.201894591845071 0.0001623113145232669 0.000162324486648811 -4.52798958873942
56 1.201900286087054 0.0001566170725406657 0.0001566293366747592 -4.525464645351152
57 1.201905685859183 0.0001512173004110107 0.0001512287334593573 -4.5230091053239
58 1.201910811120572 0.000146092039022605 0.0001461027102052743 -4.520619767225281
59 1.201915680167553 0.0001412229920412233 0.0001412329637737448 -4.518293638760038
60 1.201920309797183 0.0001365933624115634 0.0001366026910730141 -4.516027905959973
61 1.201924715452282 0.0001321877073126032 0.0001321964439156587 -4.513819932780932
62 1.201928911350372 0.0001279918092225163 0.000128 -4.511667243064355
63 1.201932910598513 0.0001239925610811987 0.000124000248000496 -4.50956750844126
64 1.201936725295779 0.0001201778638155737 0.0001201850850309477 -4.507518530365488
65 1.201940366624864 0.000116536534730427 0.0001165433249810617 -4.505518239614971
66 1.201943844934127 0.0001130582254669221 0.0001130646164282888 -4.503564683263746
67 1.20194716981119 0.0001097333484043617 0.0001097393689986283 -4.501656025894788
68 1.201950350149069 0.0001065530105253121 0.0001065586871969737 -4.499790515309039
69 1.201953394205701 0.0001035089538936607 0.0001035143108534755 -4.497966501510966
70 1.201956309657596 0.000100593501998647 0.0001005985614405714 -4.496182427673079
71 1.20195910364828 9.779951131383413e-05 9.780429360848941e-05 -4.49443681084581
72 1.201961782832094 9.512032750036425e-05 9.512485136741974e-05 -4.492728244598787
73 1.201964353413842 9.254974575201302e-05 9.255402841408673e-05 -4.491055399816985
74 1.201966821184754 9.008197484017266e-05 9.008603216071348e-05 -4.489417011148479
75 1.201969191555124 8.771160446974235e-05 8.771545107670716e-05 -4.487811865496496
76 1.201971469583992 8.543357560264475e-05 8.543722499893203e-05 -4.486238832357404
77 1.201973660006152 8.324315344254707e-05 8.324661810613944e-05 -4.484696804868141
78 1.20197576725678 8.113590281455529e-05 8.113919428780072e-05 -4.483184751247111
79 1.201977795493897 7.910766569740879e-05 7.911079466793244e-05 -4.481701682655552
80 1.201979748618897 7.715454069745498e-05 7.715751707110065e-05 -4.480246655628918
81 1.20198163029532 7.527286427433388e-05 7.527569724114569e-05 -4.478818763916544
82 1.201983443966044 7.345919354984076e-05 7.346189164370983e-05 -4.477417139728746
83 1.201985192869045 7.171029054919842e-05 7.171286170174621e-05 -4.476040964568041
84 1.201986880051855 7.002310773951415e-05 7.002555932915514e-05 -4.474689451142798
85 1.201988508384849 6.839477474551714e-05 6.839711364180432e-05 -4.47336186065972
86 1.201990080573461 6.682258613310843e-05 6.682481873767917e-05 -4.47205743944392
87 1.201991599169428 6.530399016613231e-05 6.530612244897959e-05 -4.470775515614187
88 1.201993066581149 6.383657844555302e-05 6.383861597880558e-05 -4.469515423981185
89 1.201994485083239 6.241807635531949e-05 6.24200243438095e-05 -4.468276529506308
90 1.201995856825351 6.104633424275363e-05 6.104819755196727e-05 -4.46705822147225
91 1.201997183840324 5.971931926995744e-05 5.972110245155126e-05 -4.465859915788036
92 1.201998468051716 5.843510787850725e-05 5.843681519357195e-05 -4.464681064582156
93 1.201999711280779 5.719187881481069e-05 5.719351425548342e-05 -4.463521115171593
94 1.202000915252924 5.598790666994091e-05 5.598947397889197e-05 -4.462379551348827
95 1.202002081603704 5.482155588998339e-05 5.482305857843809e-05 -4.46125587352416
96 1.202003211884376 5.369127521870354e-05 5.369271658299552e-05 -4.460149597033682
97 1.202004307567057 5.259559253723012e-05 5.259697567389875e-05 -4.45906026950906
98 1.202005370049526 5.153311006811379e-05 5.153443788811873e-05 -4.457987418363708
99 1.202006400659678 5.050249991600531e-05 5.0503775157193e-05 -4.456930651941094
100 1.202007400659678 4.950249991608757e-05 4.950372515531794e-05 -4.455889553084795
=-=-=-=-=-=-=-=
EEEDDDDIIITTTT:
Do the same trick one more time, $$ \sum_{k = 1}^n \frac{1}{k^3} +\frac{2}{(2n+1)^2} - \frac{2}{(2n+1)^4} = \zeta(3) - O(\frac{1}{n^6}) $$
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
est = 2 / (2 n + 1)^2 - 2 / (2 n + 1)^4
const double zeta_3 = 1.20205690315959428539 ;
n sump est sump + est
1 1 0.1975308641975309 1.197530864197531
2 1.125 0.07680000000000001 1.2018
3 1.162037037037037 0.03998334027488546 1.202020377311922
4 1.177662037037037 0.02438652644413961 1.202048563481177
5 1.185662037037037 0.0163923229287617 1.202054359965799
6 1.190291666666667 0.01176429396729806 1.202055960633965
7 1.19320711856171 0.008849382716049382 1.20205650127776
8 1.19516024356171 0.006896469151470888 1.202056712713181
9 1.196531985674193 0.005524819484196714 1.20205680515839
10 1.197531985674193 0.004524863611355351 1.202056849285548
11 1.198283300475095 0.003773571420914019 1.202056871896009
12 1.198862004178798 0.00319488 1.202056884178798
13 1.199317170314438 0.002739720872119389 1.202056891186557
14 1.199681601801318 0.002375293553764345 1.202056895355083
15 1.199977898097615 0.002078999827832827 1.202056897925447
16 1.200222038722615 0.001834860838116536 1.202056899560731
17 1.200425580346877 0.001631320283215327 1.202056900630092
18 1.200597048110937 0.001459853235661184 1.202056901346598
19 1.200742841958437 0.001314059878769331 1.202056901837206
20 1.200867841958436 0.001189060221299678 1.202056902179736
21 1.200975821658253 0.001081080764864641 1.202056902423118
22 1.201069736008366 0.0009871665904587715 1.202056902598824
23 1.201151925537419 0.0009049771901073041 1.202056902727527
24 1.201224263500382 0.0008326393226756656 1.202056902823058
25 1.201288263500383 0.0007686393944540598 1.202056902894837
26 1.201345159267337 0.0007117436820391559 1.202056902949377
27 1.201395964530763 0.0006609384604876717 1.20205690299125
28 1.201441518466623 0.0006153845570874804 1.20205690302371
29 1.20148252055773 0.0005743824913692016 1.202056903049099
30 1.201519557594767 0.0005373454743558011 1.202056903069122
31 1.201553124779487 0.0005037783055515722 1.202056903085038
32 1.201583642357612 0.0004732607401701621 1.202056903097782
33 1.201611468831719 0.0004454342763362892 1.202056903108055
34 1.201636911534752 0.0004199915816393268 1.202056903116391
35 1.201660235149912 0.0003966679732836249 1.202056903123196
36 1.20168166862042 0.0003752345083626834 1.202056903128783
37 1.201701410787715 0.0003554923456790124 1.202056903133394
38 1.201719635018653 0.0003372681185676084 1.20205690313722
39 1.201736493023676 0.0003204101167337752 1.20205690314041
40 1.201752118023676 0.0003047851194054944 1.202056903143082
41 1.201766627389472 0.0002902757558574396 1.202056903145329
42 1.201780124851949 0.0002767782952790316 1.202056903147228
43 1.201792702360848 0.0002642007879907197 1.202056903148838
44 1.201804441654612 0.0002524614955975193 1.202056903150209
45 1.201815415591512 0.0002414875598689732 1.202056903151381
46 1.201825689282644 0.0002312138697414903 1.202056903152385
47 1.201835321059803 0.0002215820934461829 1.202056903153249
48 1.201844363305173 0.0002125398488213182 1.202056903153994
49 1.201852863164925 0.0002040399897143025 1.20205690315464
50 1.201860863164925 0.0001960399902744945 1.2020569031552
51 1.201868401743602 0.0001885014120857926 1.202056903155688
52 1.201875513714471 0.0001813894416421141 1.202056903156113
53 1.201882230668736 0.0001746724877504014 1.202056903156486
54 1.201888581326664 0.0001683218301490907 1.202056903156813
55 1.201894591845071 0.0001623113120293281 1.2020569031571
56 1.201900286087054 0.0001566170703002056 1.202056903157354
57 1.201905685859183 0.0001512172983944454 1.202056903157578
58 1.201910811120572 0.0001460920372043097 1.202056903157776
59 1.201915680167553 0.0001412229903987166 1.202056903157952
60 1.201920309797183 0.0001365933609254099 1.202056903158108
61 1.201924715452282 0.0001321877059657667 1.202056903158247
62 1.201928911350372 0.000127991808 1.202056903158372
63 1.201932910598513 0.0001239925599697439 1.202056903158483
64 1.201936725295779 0.0001201778628036157 1.202056903158582
65 1.201940366624864 0.0001165365338077629 1.202056903158672
66 1.201943844934127 0.0001130582246245447 1.202056903158752
67 1.20194716981119 0.0001097333476340741 1.202056903158824
68 1.201950350149069 0.0001065530098200652 1.202056903158889
69 1.201953394205701 0.0001035089532471997 1.202056903158948
70 1.201956309657596 0.0001005935014052894 1.202056903159001
71 1.20195910364828 9.779951076856528e-05 1.202056903159049
72 1.201961782832094 9.51203269987459e-05 1.202056903159093
73 1.201964353413842 9.254974528999889e-05 1.202056903159132
74 1.201966821184754 9.008197441411825e-05 1.202056903159168
75 1.201969191555124 8.771160407652837e-05 1.202056903159201
76 1.201971469583992 8.543357523922428e-05 1.202056903159231
77 1.201973660006152 8.324315310642638e-05 1.202056903159258
78 1.20197576725678 8.113590250337588e-05 1.202056903159283
79 1.201977795493897 7.910766540901595e-05 1.202056903159306
80 1.201979748618897 7.715454042988036e-05 1.202056903159327
81 1.20198163029532 7.527286402584812e-05 1.202056903159346
82 1.201983443966044 7.345919331894789e-05 1.202056903159363
83 1.201985192869045 7.171029033447948e-05 1.20205690315938
84 1.201986880051855 7.002310753967546e-05 1.202056903159394
85 1.201988508384849 6.839477455922706e-05 1.202056903159408
86 1.201990080573461 6.682258595947951e-05 1.202056903159421
87 1.201991599169428 6.530399000416493e-05 1.202056903159432
88 1.201993066581149 6.383657829436054e-05 1.202056903159443
89 1.201994485083239 6.241807621408995e-05 1.202056903159453
90 1.201995856825351 6.104633411075511e-05 1.202056903159462
91 1.201997183840324 5.971931914651225e-05 1.202056903159471
92 1.201998468051716 5.843510776288696e-05 1.202056903159479
93 1.201999711280779 5.719187870644698e-05 1.202056903159486
94 1.202000915252924 5.598790656829375e-05 1.202056903159493
95 1.202002081603704 5.482155579456214e-05 1.202056903159499
96 1.202003211884376 5.369127512908849e-05 1.202056903159505
97 1.202004307567057 5.259559245297373e-05 1.20205690315951
98 1.202005370049526 5.153310998897451e-05 1.202056903159515
99 1.202006400659678 5.050249984154044e-05 1.20205690315952
100 1.202007400659678 4.950249984591581e-05 1.202056903159524
101 1.202008371249826 4.853190970206915e-05 1.202056903159528
102 1.202009313572161 4.758958737133113e-05 1.202056903159532
103 1.20201022871382 4.667444571553553e-05 1.202056903159536
104 1.202011117710179 4.578544936018951e-05 1.202056903159539
105 1.202011981547777 4.492161176476751e-05 1.202056903159542
106 1.20201282116706 4.408199248464505e-05 1.202056903159545
107 1.202013637464937 4.326569461047894e-05 1.202056903159548
108 1.202014431297178 4.247186237201173e-05 1.20205690315955
109 1.202015203480658 4.169967889434411e-05 1.202056903159553
110 1.202015954795459 4.094836409568776e-05 1.202056903159555
111 1.20201668598684 4.021717271649421e-05 1.202056903159557
112 1.202017397767088 3.950539247065996e-05 1.202056903159559
113 1.202018090817251 3.881234231024203e-05 1.202056903159561
114 1.202018765788767 3.813737079578854e-05 1.202056903159563
115 1.202019423304999 3.747985456500138e-05 1.202056903159564
116 1.202020063962673 3.683919689300814e-05 1.202056903159566
117 1.202020688333229 3.621482633803317e-05 1.202056903159567
118 1.202021296964102 3.560619546672738e-05 1.202056903159568
119 1.202021890379916 3.50127796538467e-05 1.20205690315957
120 1.20202246908362 3.443407595136405e-05 1.202056903159571
121 1.20202303355755 3.386960202246241e-05 1.202056903159572
122 1.202023584264437 3.331889513618944e-05 1.202056903159573
123 1.202024121648355 3.278151121886033e-05 1.202056903159574
124 1.202024646135617 3.225702395857755e-05 1.202056903159575
125 1.202025158135617 3.174502395949511e-05 1.202056903159576
126 1.202025658041634 3.12451179426942e-05 1.202056903159577
127 1.202026146231587 3.075692799075741e-05 1.202056903159578
128 1.202026623068745 3.028009083333148e-05 1.202056903159578
129 1.202027088902408 2.981425717115682e-05 1.202056903159579
130 1.202027544068544 2.935909103621458e-05 1.20205690315958
131 1.202027988890395 2.891426918580266e-05 1.202056903159581
132 1.202028423679053 2.84794805285001e-05 1.202056903159581
133 1.202028848734002 2.805442558011591e-05 1.202056903159582
134 1.202029264343635 2.763881594784607e-05 1.202056903159583
135 1.202029670785742 2.723237384097912e-05 1.202056903159583
136 1.202030068327977 2.683483160660051e-05 1.202056903159584
137 1.202030457228295 2.644593128884639e-05 1.202056903159584
138 1.202030837735374 2.606542421035173e-05 1.202056903159584
139 1.20203121008901 2.569307057462455e-05 1.202056903159585
140 1.202031574520497 2.532863908815914e-05 1.202056903159585
141 1.202031931252985 2.497190660117611e-05 1.202056903159586
142 1.20203228050182 2.462265776594754e-05 1.202056903159586
143 1.202032622474875 2.428068471173021e-05 1.202056903159586
144 1.202032957372851 2.394578673539059e-05 1.202056903159587
145 1.20203328538958 2.361777000686207e-05 1.202056903159587
146 1.202033606712299 2.329644728862704e-05 1.202056903159588
147 1.202033921521919 2.298163766846593e-05 1.202056903159588
148 1.202034229993283 2.267316630476089e-05 1.202056903159588
149 1.202034532295405 2.237086418368455e-05 1.202056903159588
150 1.202034828591701 2.207456788764433e-05 1.202056903159589
151 1.202035119040215 2.178411937438973e-05 1.202056903159589
152 1.202035403793823 2.14993657662254e-05 1.202056903159589
153 1.202035683000441 2.122015914880475e-05 1.202056903159589
154 1.202035956803211 2.094635637900987e-05 1.20205690315959
155 1.202036225340688 2.067781890145168e-05 1.20205690315959
156 1.202036488747017 2.041441257315118e-05 1.20205690315959
157 1.202036747152094 2.015600749598758e-05 1.20205690315959
158 1.202037000681734 1.990247785652246e-05 1.20205690315959
159 1.202037249457818 1.965370177283133e-05 1.202056903159591
160 1.202037493598443 1.940956114799432e-05 1.202056903159591
161 1.202037733218061 1.916994152991712e-05 1.202056903159591
162 1.202037968427614 1.893473197717167e-05 1.202056903159591
163 1.202038199334661 1.87038249305628e-05 1.202056903159592
164 1.202038426043502 1.847711609014333e-05 1.202056903159592
165 1.202038648655294 1.825450429741501e-05 1.202056903159592
166 1.202038867268169 1.803589142246686e-05 1.202056903159592
167 1.202039081977336 1.782118225581594e-05 1.202056903159592
168 1.202039292875187 1.76102844047278e-05 1.202056903159592
169 1.202039500051398 1.740310819380592e-05 1.202056903159592
170 1.202039703593023 1.719956656965038e-05 1.202056903159592
171 1.202039903584583 1.699957500939656e-05 1.202056903159592
172 1.20204010010816 1.680305143295437e-05 1.202056903159592
173 1.202040293243474 1.660991611877795e-05 1.202056903159592
174 1.20204048306797 1.642009162300445e-05 1.202056903159593
175 1.202040669656891 1.623350270180869e-05 1.202056903159593
176 1.202040853083356 1.605007623682847e-05 1.202056903159593
177 1.202041033418429 1.58697411635224e-05 1.202056903159593
178 1.202041210731191 1.569242840232932e-05 1.202056903159593
179 1.202041385088801 1.551807079250472e-05 1.202056903159593
180 1.202041556556565 1.534660302851582e-05 1.202056903159593
181 1.202041725197994 1.5177961598883e-05 1.202056903159593
182 1.202041891074866 1.50120847273604e-05 1.202056903159593
183 1.202042054247277 1.484891231635424e-05 1.202056903159593
184 1.202042214773701 1.468838589248193e-05 1.202056903159593
185 1.202042372711039 1.453044855417991e-05 1.202056903159593
186 1.202042528114672 1.437504492127275e-05 1.202056903159594
187 1.202042681038507 1.422212108641975e-05 1.202056903159594
188 1.202042831535025 1.407162456835991e-05 1.202056903159594
189 1.202042979655327 1.392350426687923e-05 1.202056903159594
190 1.202043125449174 1.377771041942854e-05 1.202056903159594
191 1.202043268965034 1.363419455932274e-05 1.202056903159594
192 1.202043410250118 1.349290947545619e-05 1.202056903159594
193 1.20204354935042 1.335380917347157e-05 1.202056903159594
194 1.202043686310756 1.321684883832263e-05 1.202056903159594
195 1.202043821174796 1.308198479817395e-05 1.202056903159594
196 1.202043953985104 1.294917448958342e-05 1.202056903159594
197 1.20204408478317 1.281837642391552e-05 1.202056903159594
198 1.202044213609439 1.268955015493615e-05 1.202056903159594
199 1.202044340503346 1.256265624754142e-05 1.202056903159594
200 1.202044465503346 1.24376562475756e-05 1.202056903159594
n sump est sump + est
const double zeta_3 = 1.20205690315959428539 ;
est = 2 / (2 n + 1)^2 - 2 / (2 n + 1)^4
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