Find $\lim_{n \to \infty} \sqrt[n]{1+2^{(-1)^n}}$ and $\lim_{n \to \infty} \sqrt[n]{1^k+2^k+\cdots+n^k}$

Question

Here is a problem I haven't met with yet and looks curious. Find $$\lim_{n \to \infty} \sqrt[n]{1+2^{(-1)^n}}$$ Also help with this one would be appreciated $$\lim_{n \to \infty} \sqrt[n]{1^k+2^k+\cdots+n^k}$$ where k is an integer. We know that the first limit converges because it's $<1< for ratio test.

Answer 1

Try the squeeze theorem. Depending on whether $n$ is even or odd, we have respectively $$ 1+2^{(-1)^n} = \begin{cases}3&n \text{ even}\\ \frac{3}{2}&n \text{ odd} \end{cases} $$ so that $$ \left(\frac{3}{2}\right)^{1/n} \leq \left(1+2^{(-1)^n}\right)^{1/n} \leq 3^{1/n}. $$ The limit of both LHS and RHS when $n\to\infty$ is...?

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For the second one, same approach. Since $k>0$ is a constant, you have $$ n \leq \sum_{j=1}^n j^k \leq n^{k+1} $$and thus $$ n^{1/n} \leq \left(\sum_{j=1}^n j^k\right)^{1/n} \leq n^{(k+1)/n} = e^{(k+1)\frac{\ln n}{n}}. $$ Again, both LHS and RHS have same limit.