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.