I would like to know how I proof that the following infinite series is divergent:
$$\sum_{i=1}^n (-1)^n*(\frac{n}{n+1})^n$$
I tried the following tests: ratio test, root test, Leibniz criterion, divergence test, but they lead to an inconculsive result. The other only tests I am allowed to use is the comparison test and Cauchy's convergence test , but I don't know how to proof that the infinite series is divergent with these.
[Math] Prove that the infinite series is divergent
calculussequences-and-series
Best Answer
The series has the $n^{th}$ term $a_n=\dfrac{1}{\left(-1-\dfrac{1}{n}\right)^n}$ does not converge to $0$. Thus the series diverges.