Question

Determine whether the series is convergent or divergent. 1 4 + 3 16 + 1 64 + 3 256 + 1 1024 + 3 4096 +

Answer 1

This series converges.

$S=\dfrac14+\dfrac3{16}+\dfrac1{64}+\dfrac3{256}+\dfrac1{1024}+\cdots$

$S=\left(\dfrac34+\dfrac3{16}+\dfrac3{64}+\dfrac3{256}+\dfrac3{1024}+\cdots\right)-\left(\dfrac12+\dfrac1{32}+\dfrac1{512}+\cdots\right)$

$S=\displaystyle\sum_{n=1}^\infty\frac3{4^n}-\sum_{n=0}^\infty\frac1{2^{4n+1}}$

Both component series are geometric with ratios less than 1, so they both converge.

$\displaystyle\sum_{n=1}^\infty\frac3{4^n}=1$

$\displaystyle\sum_{n=0}^\infty\frac1{2^{4n+1}}=\frac12\sum_{n=0}^\infty\frac1{16^n}=\frac8{15}$

So we have

$S=\dfrac7{15}$