Question

Find the sum to infinity of the series 2+5/4+11/16+23/64+..........up to the infinity.

infinity​

Answer 1

We have

`2+\frac54+(11)/(16)+(23)/(64)+\cdots=\displaystyle\sum_(n=0)^\infty(3\cdot2^n-1)/(4^n)`

(notice that each denominator is a power of 4, and each numerator is one less than some multiple of 3, in particular 3 times some power of 2)

Recall for
`|x|<1`, we have

`\displaystyle\frac1{1-x}=\sum_(n=0)^\infty x^n`

So we have

`\displaystyle\sum_(n=0)^\infty\frac{3\cdot2^n-1}4=3\sum_(n=0)^\infty\left(\frac12\right)^n-\sum_(n=0)^\infty\left(\frac14\right)^n=\frac3{1-\frac12}-\frac1{1-\frac14}=\boxed{\frac{14}3}`