Prove that $\left ( 1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}<2$

Note that by Bernoulli inequality in the form

$$(1+x)^a<1+ax, \quad 0<a<1$$

which can be easily proved by induction, we obtain

$$\left ( 1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}<1+\frac{n^{\frac{1}{n}}}{n^2}+1-\frac{n^{\frac{1}{n}}}{n^2} =2$$

By the generalized binomial formula, for $x\ne0$,

$$(1+x)^{1/n}+(1-x)^{1/n} \\=2+\frac2n\left(\frac1n-1\right)\frac{x^2}2+\frac2n\left(\frac1n-1\right)\left(\frac1n-2\right)\left(\frac1n-3\right)\frac{x^4}{4!}+\cdots\\<2.$$

That's all folks.

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Alternatively, the first derivative of $(1+x)^{1/n}+(1-x)^{1/n}$ is odd and monotonic in $(-1,1)$ because the second derivative is non-negative, so it has a single maximum, at $(0,2)$.

Since $f(x)=x^{\frac{1}{n}}$ is a concave function for $n>1$, by Jensen we obtain: $$\left (1+\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}+\left ( 1-\frac{n^{\frac{1}{n}}}{n} \right )^\frac{1}{n}\leq2\left (\frac{1+\frac{n^{\frac{1}{n}}}{n}+ 1-\frac{n^{\frac{1}{n}}}{n}}{2}\right )^\frac{1}{n}=2.$$ The equality occurs for $$1+\frac{n^{\frac{1}{n}}}{n}=1-\frac{n^{\frac{1}{n}}}{n},$$ which says that the equality does not occur.