Geometric series with even and odd coefficient

Question

I have to determine the limit of a following geometric series:

$$t_n=\sum_{k=0}^n \frac{a_k}{2^k} $$ in which $$a_k=\begin{cases} 3 & \text{if $k$ is even}\\[2ex] 4 & \text{if $k$ is odd} \end{cases} $$

I'm not actually sure how should I approach this particular problem. Shall I examine the two cases separately and determine the limit for both cases respectively? Or is there some kind of trick to tackle this kind of a geometric series?

Answer 1

We have $$ \sum_{j=0}^{+\infty}\frac{3}{2^{2j}}=4,\qquad \sum_{j=0}^{+\infty}\frac{4}{2^{2j+1}}=\frac{8}{3}$$ (both are geometric series) hence: $$ \sum_{k\geq 0}\frac{a_k}{2^k} = \color{red}{\frac{20}{3}}.$$ The rearrangement/splitting is allowed since we are clearly dealing with an absolutely convergent series.

Answer 2

$$t_n=\begin{cases} \sum_{k=0}^n \frac{3}{2^k} & \text{if $k$ is even}\\[2ex] \sum_{k=0}^n \frac{4}{2^k} & \text{if $k$ is odd} \end{cases} $$

Consider the two subsequences $t_{2n}$ and $t_{2n-1}$ where
$t_{2n}=\sum_{k=0}^{2n} \frac{3}{2^k}$
and $t_{2n-1}=\sum_{k=0}^{2n-1} \frac{4}{2^k}$

Now these subsequences include all terms of the main series and they are both monotonic decreasing and bounded below.

If they converge to same limit, they converge. Else, they do not.

Now do the subsequences have the same limit? Check it.

Answer 3

$t_n=\sum_{k=0}^n \frac{a_k}{2^k}=3(\frac{1}{2^0}+\frac{1}{2^2}+\cdot\cdot\cdot)+4(\frac12+\frac{1}{2^3}+\cdot\cdot\cdot)=(3+\frac42)(\frac{1}{2^0}+\frac{1}{2^2}+\cdot\cdot\cdot)=5(\frac{1}{1-\frac14})$

Thus $$\sum_{k=0}^{\infty} \frac{a_k}{2^k}=\frac{20}{3}$$