Question

Find the sum of the infinite series

Question is in the attachment ​

Answer 1

Answer:

$\boxed{\huge{\mathbb\pink{REFERR \:TO\: THE\:\: ATTACHMENT }}}$

Step-by-step explanation:

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Answer 2

Given Question

Find the sum of

$\displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{n + k} }$

$\red{\large\underline{\sf{Solution-}}}$

Given series is

$\rm :\longmapsto\:\displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{n + k} }$

can be further rewritten as

$\rm \:  =  \: \:\displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{k} . {2}^{n} }$

can be further rewritten as

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \bigg[\dfrac{k}{ {2}^{k}}\displaystyle\sum_{n=1}^{\infty}\rm \dfrac{1}{ {2}^{n} } \bigg]$

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \dfrac{k}{ {2}^{k} }\bigg(\dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + \dfrac{1}{ {2}^{3} } - - - \infty \bigg)$

So, its an infinite GP series with

$\purple{\rm :\longmapsto\:a = \dfrac{1}{2}}$

$\purple{\rm :\longmapsto\:r = \dfrac{1}{2}}$

We know,

Sum of infinite GP series with common ratio r ( - 1 < r < 1 ) and first term a is given by

$\boxed{\tt{ \: \: S_ \infty = \frac{a}{1 - r} , \: \: provided \: that \: |r| < 1}}$

So, using this, we get

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{1 - \frac{1}{2} }\bigg)$

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{ \frac{2 - 1}{2} }\bigg)$

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }\bigg( \frac{ \frac{1}{2} }{ \frac{1}{2} }\bigg)$

$\rm \:  =  \: \displaystyle\sum_{k=1}^{\infty}\rm \frac{k}{ {2}^{k} }$

$\rm \:  =  \: \dfrac{1}{2} + \dfrac{2}{ {2}^{2} } + \dfrac{3}{ {2}^{3} } + \dfrac{4}{ {2}^{4} } + - - - \infty$

Let assume that

$\rm :\longmapsto\:S_ \infty   =  \: \dfrac{1}{2} + \dfrac{2}{ {2}^{2} } + \dfrac{3}{ {2}^{3} } + \dfrac{4}{ {2}^{4} } + - - - \infty$

Now, its an infinite Arithmetico Geometrico Series, So multiply by 1/2 we get

$\rm :\longmapsto\:\dfrac{1}{2} S_ \infty   =  \: \dfrac{1}{ {2}^{2} } + \dfrac{2}{ {2}^{3} } + \dfrac{3}{ {2}^{4} } + \dfrac{4}{ {2}^{5} } + - - - \infty$

On Subtracting above two equations, we get

$\rm :\longmapsto\:\dfrac{1}{2} S_ \infty   =  \: \dfrac{1}{2} + \dfrac{1}{ {2}^{2} } + \dfrac{1}{ {2}^{3} } + \dfrac{1}{ {2}^{4} } + - - - \infty$

$\rm :\longmapsto\:\dfrac{1}{2} S_ \infty   =  \: \dfrac{ \dfrac{1}{2} }{1 - \dfrac{1}{2} }$

$\rm :\longmapsto\:\dfrac{1}{2} S_ \infty   =  \: \dfrac{ \dfrac{1}{2} }{ \dfrac{1}{2} }$

$\rm :\longmapsto\:\dfrac{1}{2} S_ \infty   =  \: 1$

$\bf\implies \:S_ \infty = 2$

Hence,

$\\ \rm :\longmapsto\:\boxed{\tt{ \: \: \: \: \displaystyle\sum_{k=1}^{\infty}\rm \displaystyle\sum_{n=1}^{\infty}\rm \frac{k}{ {2}^{n + k} } = 2 \: \: \: \: }}$