Putnam and Beyond problem 321

Question

If $\lim_{x\to0^+} x^x=1$ then is $\lim_{x\to0^+} x^{x+1}=0$? Similarly can we conclude $\lim_{n\to\infty} (k/n^2)^{k/n^2 +1}=0$? If not, why? Edit: Why is $\lim_{n\to\infty} \sum\limits_{k=1}^n (k/n^2)^{k/n^2 +1}=1/2$ but not 0?

Answer 1

Recall that for any $x_n\to x_0$

$$\lim_{x\to x_0} f(x)=L \implies \lim_{n\to \infty} f(x_n)=L$$

Refer to Proving that a subsequence of a function converges to the same limit.

Note that also $\frac 1n \to 0$ but $\sum \frac 1n=\infty$.

Indeed if the limit of $a_n \to 0$ with $a_i > 0$ then $\sum a_n$ can't be equal to zero.