4. Consider the series $\sum a_n$ where $$a_n=\begin{cases}n/2^n&n\text{ odd}\\1/2^n&n\text{ even}\end{cases}$$
- a. Show the Ratio Test is inconclusive
- b. Use the Root Test to determine whether the series is convergent or divergent.
Not sure how to do this..
Best Answer
Hints:
The ratio test gives
$$\frac{a_{n+1}}{a_n}=\begin{cases}\cfrac{\frac{n+1}{2^{n+1}}}{\frac1{2^n}}=\frac{n+1}2\,,&n\;\text{ is even}\\{}\\\cfrac{\frac1{2^{n+1}}}{\frac n{2^n}}=\frac1{2n}\,,&n\;\text{ is odd}\end{cases}$$
and thus the limit doesn't exist so...
For the $\;n\,-$ th root test:
$$\sqrt[n]{a_n}=\begin{cases}\cfrac{\sqrt[n]n}2\,,&n\;\text{ is odd}\\{}\\\;\;\;\cfrac12\,,&n\;\text{ is even}\end{cases}\;\xrightarrow[n\to\infty]{}\ldots$$
and thus the limit exists and...