We have to determin the convergence of the following sum:
\begin{equation}
\sum\limits_{n=1}^{\infty}\frac{-(n-1)}{n\sqrt{n+1}}
\end{equation}
And we've seen multiple ways to determine if the sum converges or diverges, but they only work with non negative terms. So tests like d'Alemberts ratio test, Cauchy's root test or Raabe's test wont work. How would I go about determining the convergence of this sum?
Best Answer
For $n>2$, $$\frac{n-1}{n\sqrt{n+1}}>\frac1{2\sqrt{n+1}} $$ As $\sum \frac1{2\sqrt{n+1}}$ diverges. so does $\sum\frac{n-1}{n\sqrt{n+1}}$ and also $\sum\frac{-(n-1)}{n\sqrt{n+1}}$.