<s>[INST] Show that a ratio test is inconclusive for a given series, then determine whether the series converges/diverges? 
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..
 [/INST] The series is a combination of geometric series and Gabriel's Staircase:
$$\sum_{k=0}^\infty a_k = \sum_{k=0}^\infty a_{2k}+\sum_{k=0}^\infty a_{2k+1} =  \sum_{k=0}^\infty (\frac{1}{2})^{2k} + \sum_{k=0}^\infty (2k+1)(\frac{1}{2})^{2k+1}$$
We split again, rearrange and combine again:
$$=  \sum_{k=0}^\infty (\frac{1}{2})^{2k} + \sum_{k=0}^\infty k (\frac{1}{4})^{k} + \sum_{k=0}^\infty (\frac{1}{2})^{2k+1} = \sum_{k=0}^\infty (\frac{1}{2})^{k} + \sum_{k=0}^\infty k (\frac{1}{4})^{k}$$
Now both series converge and we get as the combined value:
$$= \frac{1}{1-\frac{1}{2}} + \frac{1}{4} \frac{1}{(1-\frac{1}{4})^2} = 2 + \frac{1}{4} \frac{16}{9} = \frac{22}{9}$$
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