Question

The limiting sum of the infinite series, $\dfrac{1}{{10}} + \dfrac{2}{{{{10}^2}}} + \dfrac{3}{{{{10}^3}}} + ..................$whose ${n^{th}}$term is $\dfrac{n}{{{{10}^n}}}$is
${\text{A}}{\text{. }}\dfrac{1}{9} \\\ {\text{B}}{\text{. }}\dfrac{{10}}{{81}} \\\ {\text{C}}{\text{. }}\dfrac{1}{8} \\\ {\text{D}}{\text{. }}\dfrac{{17}}{{72}} \\\$

Answer 1

Hint:- In this question, given series is in Arithmetic –Geometric Progression(AGP) so we first convert it into Geometric Progression(GP) and then apply infinite series summation of GP($Su{m_\infty } = \dfrac{a}{{1 - r}}$) to get answer.

Complete step-by-step solution -

Arithmetic –Geometric Progression(AGP) is a progression in which each term can be represented as the product of the terms of an Arithmetic Progression(AP) and a Geometric Progression(GP).
Let $S = \dfrac{1}{{10}} + \dfrac{2}{{{{10}^2}}} + \dfrac{3}{{{{10}^3}}} + ..................$ eq.1
Since, in S numerator varies in AP while denominator varies in GP. Hence, it is in AGP.
Now, divide eq.1 with 10, we get
$\Rightarrow \dfrac{S}{{10}} = \dfrac{1}{{{{10}^2}}} + \dfrac{2}{{{{10}^3}}} + \dfrac{3}{{{{10}^4}}}..................{\text{ eq}}{\text{.2}}$
Subtract eq.2 from eq.1, we get
$\Rightarrow \dfrac{{9S}}{{10}} = \dfrac{1}{{{{10}^1}}} + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^3}}} + \dfrac{1}{{{{10}^4}}}..................{\text{ eq}}{\text{.3}}$
Multiple eq.3 with 10, we get
$\Rightarrow 9S = 1 + \dfrac{1}{{{{10}^1}}} + \dfrac{1}{{{{10}^2}}} + \dfrac{1}{{{{10}^3}}} + \dfrac{1}{{{{10}^4}}}..................{\text{ eq}}{\text{.4}}$
We can observe that RHS of eq.4 is in Geometric Progression(GP). We know that sum of infinite GP series is given by
$\Rightarrow Su{m_\infty } = \dfrac{a}{{1 - r}}$
Where $a = {\text{first term of GP}} \\\ r = {\text{ common ratio}} \\\$
From eq.4 we can observe that $a = 1{\text{ and }}r = \dfrac{1}{{10}}$
Then, $\Rightarrow 9{S_\infty } = \dfrac{1}{{1 - \dfrac{1}{{10}}}} \\\ \Rightarrow 9{S_\infty } = {\text{ }}\dfrac{{10}}{9} \\\ \Rightarrow {S_\infty } = {\text{ }}\dfrac{{10}}{{81}} \\\$
Hence, option B. is correct.
Note:- Whenever we get this type of question first we need to observe that the given series is in AGP or AP or GP. If it is in AGP we convert it into GP by performing simple operations. Later we have to apply the concept of AGP and infinite series summation formula of GP(${S_\infty } = \dfrac{a}{{1 - r}}$).