Question

is this serie convergent? sigma [(n-1)/(n+1)]^n by which tests ?

Answer 1

Hmm...I think the series diverges...lemme write it out..

Answer 2

Trying the n'th term test, let's directly observe the stuff within the ()^n
So we have limit as n approaches infinity of (n-1)/(n+1) which is 1.
By intuition, we can see that limit as n->infinity [(n-1)/(n+1)]^n is not = 0.
Hence the series diverges.

Answer 3

Given a series:
\[\sum_k a_k\]
You know it DIVERGES (DOES NOT CONVERGE) if:
\[\lim_{k \rightarrow \infty} a_k = 0\]
This does NOT mean it converges, but it is a necessary condition for convergence. Therefore we have:
\[\lim_{k \rightarrow \infty} \left( \frac{k-1}{k+1} \right)^k\]
is indeterminate:
\[\left(\frac{\infty}{\infty} \right)^{\infty}\]
But 'eyeballing' it will show that the inside goes to 1 and an infinite product of 1's is 1 therefore it must diverge.

Answer 4

thanks ..