Chapter 5.6, Problem 375E

Let $a_n=2^{-\left[n/2\right]}$ where $[x]$ is the greatest integer

less than or equal to x. Determine whether ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{a}_n}$

converges and justify your answer.

The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine

their convergence. The test states that if $\underset{n\to \infty }{\lim }\frac{a_{2n}}{a_n}<1/2$ .

then ${\displaystyle \sum a_n}$ converges, while if $\underset{n\to \infty }{\lim }\frac{a_{2n+1}}{a_n}>1/2$

then a diverges.