Series converge/ diverge?
Determine whether the given infinite series converges or diverges. If it converges, find its sum.
The given series is:
1 + 1/(2^(1/2)) + 1/(3^(1/3)) + ... + 1/(n^(1/n)) + ...
I think I have to prove it converges or diverges using one of those series tests, but am not sure. And how do I find the sum if it does converge?
Thanks
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Your typical term is:
a_k = 1/(k^(1/k)) = exp[-ln(k)/k]
However, ln(k)/k => 0 as k => infinity, so a_k => 1
The sum cannot converge, because you will be summing up an infinite number of 1's .
The general term of the series tens to 1
This shows that the series is divergent since, in order to be convergent, the general term should converge to 0.