Convergence of nth root of sum of nth potence of positive real numbers

Asked Nov 10 '19 at 17:11

Active Nov 10 '19 at 17:11

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Let $k \in N$ and $a_1, a_2, ..., a_k > 0$. Prove that the following series $(b_n)$ converges and define its limit.

$$b_n := \sqrt[n]{\sum_{i = 1}^ka_i^n}$$

I'm quite sure it converges to something between $min(a_i)$ and $max(a_i)$, but don't really know where to start. Sandwich theorem wasn't of much help as well.

Thanks in advance!

asked Nov 10 '19 at 17:11

D. Petrov

Do you mean to say $a_k$ is also a positive infinite sequence, because as of right now, your question reads that $a_k$ is the last value for some fixed $k$? – gt6989b Nov 10 '19 at 17:14
Possible duplicate of Finding the limit of $\sqrt[n]{\sum_{i=1}^m |x_i|^n}$ – Martin R Nov 10 '19 at 17:14
Also: https://math.stackexchange.com/q/1188476/42969. – Martin R Nov 10 '19 at 17:15
@HansLundmark That was it, tnank you! – D. Petrov Nov 10 '19 at 17:19

Convergence of nth root of sum of nth potence of positive real numbers

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