Question

Find the infinite sum:
$\boxed{1 + \frac{2}{\pi} + \frac{3}{ {\pi}^{2} } + \frac{4}{ {\pi}^{3} } + ... + \frac{n}{ {\pi}^{n - 1} } + ... }$
Given the condition that $\lim \limits_{n \to \infty} n x^n = 0\, {\tt{when}}\, |x| < 1$​

Answer 1

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

Given series, is

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

Let assume that

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

Now, its an Arithmetico Geometrico Series, obtained by multiplying corresponding terms of following AP and GP series

$\rm \: 1,2,3,4 - - -$

and

$\rm \: 1,\dfrac{1}{\pi} ,\dfrac{2}{\pi} ,\dfrac{3}{\pi} ,\dfrac{4}{\pi} , - - -$

So,

$\rm \: Common\:ratio\:of\:GP, r \: = \: \dfrac{1}{\pi}$

Now, series is

$\rm \: S = 1 + \dfrac{2}{\pi} + \dfrac{3}{ {\pi}^{2} } + \dfrac{4}{ {\pi}^{3} } + - - - - \: \: \: \: - - (1)$

On multiply both sides by common ratio we get,

$\rm \: \dfrac{1}{\pi} S = \dfrac{1}{\pi} + \dfrac{2}{ {\pi}^{2} } + \dfrac{3}{ {\pi}^{3} } + - - - - \: \: \: \: - - (2)$

On Subtracting equation (2) from equation (1), we get

$\rm \: S - \dfrac{1}{\pi} S = 1 + \dfrac{1}{\pi} + \dfrac{1}{ {\pi}^{2} } + \dfrac{1}{ {\pi}^{3} } + - - - - \: \: \: \: - - (2)$

Now, RHS is an infinite GP series with

$\rm \: First \: term, \: a \: = \: 1$

$\rm \: Common\:ratio\:, r = \dfrac{1}{\pi}$

We know,

Sum of infinite GP series whose first term is a and Common ratio r is

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

So, using this identity, we get

$\rm \: S\bigg(1 - \dfrac{1}{\pi} \bigg) = \dfrac{1}{1 - \dfrac{1}{\pi} }$

$\rm \: S\bigg( \dfrac{\pi - 1}{\pi} \bigg) = \dfrac{1}{\dfrac{\pi - 1}{\pi} }$

$\rm \: S\bigg( \dfrac{\pi - 1}{\pi} \bigg) = \dfrac{\pi}{\pi - 1}$

$\rm\implies \:S = {\bigg(\dfrac{\pi}{\pi - 1} \bigg) }^{2}$

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ADDITIONAL INFORMATION

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\bigstar\:\:{\underline{{\boxed{\bf{{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\bigstar\:\:{\underline{{\boxed{\bf{{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference.