Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $

Let $k\geqslant1$. Since the function $x\mapsto x^{1/4}$ is increasing, $(k-1)^{1/4}\leqslant x^{1/4}\leqslant k^{1/4}$ for every $k-1\leqslant x\leqslant k$. Integrating this double inequality yields $$ (k-1)^{1/4}\leqslant\int_{k-1}^kx^{1/4}\mathrm dx\leqslant k^{1/4}. $$ Summing these from $k=1$ to $k=n$ yields $$ s_{n-1}=\sum_{k=0}^{n-1}k^{1/4}\leqslant\int_{0}^nx^{1/4}\mathrm dx\leqslant \sum_{k=1}^{n}k^{1/4}=s_n. $$ Since the integral is $\tfrac45n^{5/4}$ and $s_{n-1}=s_n-n^{1/4}$, this yields, for every $n\geqslant1$, $$ \tfrac45n^{5/4}\leqslant s_n\leqslant\tfrac45n^{5/4}+n^{1/4}. $$ Note finally that $n^{1/4}=o(n^{5/4})$ hence a (much weakened) version of this is $s_n\sim\tfrac45n^{5/4}$.

Here is a diagram to accompany Did's fine answer:

There is a standard technique that produces the complete asymptotic expansion for this sum and many others like it, which is to use harmonic sums and Mellin transforms.

Introduce the telescoping sum $$S(x) = \sum_{k\ge 1} \left(\sqrt[4]{k}-\sqrt[4]{x+k}\right).$$ This sum has the property that $$S(n) = \sum_{q=1}^n \sqrt[4]{q},$$ so that $S(n)$ is the value we are looking for.

Re-write the sum as follows: $$S(x) = \sum_{k\ge 1} \sqrt[4]{k} \left(1-\sqrt[4]{x/k+1}\right).$$

The sum term is harmonic and may be evaluated by inverting its Mellin transform.

Recall the harmonic sum identity $$\mathfrak{M}\left(\sum_{k\ge 1} \lambda_k g(\mu_k x);s\right) = \left(\sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} \right) g^*(s)$$ where $g^*(s)$ is the Mellin transform of $g(x).$

In the present case we have $$\lambda_k = \sqrt[4]{k}, \quad \mu_k = \frac{1}{k} \quad \text{and} \quad g(x) = 1 - \sqrt[4]{1+x}.$$

It follows that $$\sum_{k\ge 1} \frac{\lambda_k}{\mu_k^s} = \sum_{k\ge 1} \sqrt[4]{k}\times k^s = \zeta(-1/4-s)$$ which has fundamental strip $-1/4-s > 1$ or $s < -5/4.$

We need the Mellin transform $g^*(s)$ of $g(x)$ which is $$\int_0^\infty \left(1 - \sqrt[4]{1+x}\right) x^{s-1} dx$$ which is immediately seen to be a beta function integral with value $$g^*(s) = - \frac{1}{\Gamma(-1/4)} \Gamma(s)\Gamma(-1/4-s)$$ and fundamental strip $\langle -1, -1/4 \rangle,$ which is a problem, because the abscissa of convergence of the zeta function term lies outside this strip. Hence we need to shift $g^*(s)$ by using $$ g(x) = 1 + \frac{1}{4} x - \sqrt[4]{1+x}$$ with fundamental strip $\langle -2, -1 \rangle,$ which is good.

We have now added the following quantity to our sum: $$\sum_{k\ge 1} \sqrt[4]{k} \frac{1}{4} \frac{x}{k} = \frac{1}{4} x \zeta(3/4),$$ which we will have to remember to subtract from our final answer.

It follows that the Mellin transform $Q(s)$ of the harmonic sum $S(x)$ is given by $$Q(s) = - \frac{1}{\Gamma(-1/4)} \Gamma(s)\Gamma(-1/4-s) \zeta(-1/4-s).$$ The Mellin inversion integral here is $$\frac{1}{2\pi i} \int_{-3/2-i\infty}^{-3/2+i\infty} Q(s)/x^s ds$$ which we evaluate by shifting it to the right for an expansion at infinity.

First treat the pole from the zeta function term at $s=-5/4$, which has $$\mathrm{Res}(Q(s)/x^s; s=-5/4) = -\frac{1}{\Gamma(-1/4)} \Gamma(-5/4)\Gamma(1)\times -1 \times x^{5/4} = -\frac{4}{5} x^{5/4}.$$ For the pole at $s=-1$ we get $$\mathrm{Res}(Q(s)/x^s; s=-1) = -\frac{1}{4}\zeta(3/4)x.$$ We see that this is precisely the contribution that we added in when we shifted $g^*(s)$ into position and hence this residue will not be included in the asymptotic expansion.

For the pole at $s=-1/4$ from the compound gamma function term we obtain $$\mathrm{Res}(Q(s)/x^s; s=-1/4) = -\frac{1}{\Gamma(-1/4)} \Gamma(-1/4) \times -1 \times\zeta(0) = -\frac{1}{2} \sqrt[4]{x}.$$ For the pole at $s=0$ from the simple gamma function term we obtain $$\mathrm{Res}(Q(s)/x^s; s=0) = -\frac{1}{\Gamma(-1/4)} \Gamma(-1/4) \zeta(-1/4) = -\zeta(-1/4).$$

The remaining poles are at $s = q-1/4$ where $q\ge 1$ and contribute $$\mathrm{Res}(Q(s)/x^s; s=q-1/4) = -\frac{1}{\Gamma(-1/4)} \Gamma(q-1/4) \frac{(-1)^{q+1}}{q!} \zeta(-q) \frac{1}{x^{q-1/4}} \\ = - \prod_{p=0}^{q-1} (p-1/4) \times \frac{(-1)^{q+1}}{q!} (-1)^q \frac{B_{q+1}}{q+1} \frac{1}{x^{q-1/4}} = \frac{1}{4^q} B_{q+1} \frac{\prod_{p=0}^{q-1} (4p-1)}{(q+1)!} \frac{1}{x^{q-1/4}}.$$ The zero values of the Bernoulli numbers correctly represent cancelation of the gamma function poles by the trivial zeros of the zeta function.

Setting $x=n$ and observing that the shift to the right produces a minus sign we obtain the following asymptotic expansion: $$S(n) = \sum_{k=1}^n \sqrt[4]{k} \sim \frac{4}{5} n^{5/4} + \frac{1}{2} \sqrt[4]{n} + \zeta(-1/4) - \sum_{q\ge 1} \frac{1}{4^q} B_{q+1} \frac{\prod_{p=0}^{q-1} (4p-1)}{(q+1)!} \frac{1}{n^{q-1/4}}.$$

Actually computing the Bernoulli number terms we get the expansion $$\frac{4}{5} n^{5/4} + \frac{1}{2} \sqrt[4]{n} + \zeta(-1/4) + 1/48\,{n}^{-3/4}-{\frac {7}{15360}}\,{n}^{-11/4}+{\frac {11}{98304}}\, {n}^{-{\frac {19}{4}}}\\-{\frac {4807}{62914560}}\,{n}^{-{\frac {27}{4}} }+{\frac {13547}{134217728}}\,{n}^{-{\frac {35}{4}}}-{\frac {9360977}{ 42949672960}}\,{n}^{-{\frac {43}{4}}}\\+{\frac {191649409}{274877906944} }\,{n}^{-{\frac {51}{4}}}-{\frac {1089307862269}{351843720888320}}\,{n }^{-{\frac {59}{4}}} +\cdots.$$

This MSE link points to a series of similar calculations.

Care must be taken with the asymptotics of these expansions since the Bernoulli numbers eventually outgrow all other terms. For example when $n=10$ the sum of the first $62$ Bernoulli terms converge to produce thirty digits of precision and diverge from then on. These expansions fit the definition however, where we say that an asymptotic sequence $\{\phi_k(n)\}$ has $$|\phi_{k+1}(n)/\phi_k(n)|\rightarrow 0$$ as $n\rightarrow\infty$ and $$\lim_{n\rightarrow\infty}\frac{|S(n)-\sum_{k=1}^m\phi_k(n)|}{\phi_m(n)}=0.$$