Answer to Question #346386 in Real Analysis for Mr Alex

Question

Examine the following series for convergence: $\sum _{n=0}^{\infty }\left(\frac{n-2}{2n+3}\right)^n$

Answer 1

Use the Root Test

$\lim\limits_{n\to \infin}\sqrt[n]{|a_n|}=\lim\limits_{n\to \infin}\sqrt[n]{|(\dfrac{n-2}{2n+3})^n|}=\lim\limits_{n\to \infin}|\dfrac{n-2}{2n+3}|$

$=\lim\limits_{n\to \infin}|\dfrac{n/n-2/n}{2n/n+3/n}|=\lim\limits_{n\to \infin}|\dfrac{1-2/n}{2+3/n}|$

$=|\dfrac{1-0}{2+0}|=\dfrac{1}{2}<1$

Thus the given series converges by the Root Test.