If |a|<1 and |b|<1, then the sum of the series 1+(1+a) b+(1+a+a^2) b^2+(1+a+a^2+a^3) b^3+ …∞is e

Name: Problem 101, Chapter 10: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-10-15T16:51:33-08:00
Duration: 1 min 21 s
Channel: Manik Pulyani
Description: If |a|<1 and |b|<1, then the sum of the series 1+(1+a) b+(1+a+a^2) b^2+(1+a+a^2+a^3) b^3+ …∞is equal to (A) (1)/((1-b)(1-a b)) (B) (1)/((1-a)(1-a b)) (C) (1)/((1-a)(1-b)) (D) None of these

step 1 Here we have S is equal to 1 plus 1 plus A into B plus 1 plus A plus A square into B square plus

Question

If $|a|<1$ and $|b|<1$, then the sum of the series $1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+$ $\ldots \infty$ is equal to (A) $\frac{1}{(1-b)(1-a b)}$ (B) $\frac{1}{(1-a)(1-a b)}$ (C) $\frac{1}{(1-a)(1-b)}$ (D) None of these

   If $|a|<1$ and $|b|<1$, then the sum of the series $1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+$
$\ldots \infty$ is equal to
(A) $\frac{1}{(1-b)(1-a b)}$
(B) $\frac{1}{(1-a)(1-a b)}$
(C) $\frac{1}{(1-a)(1-b)}$
(D) None of these

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 10, Problem 101

Instant Answer

Solved by Expert Manik Pulyani

10/15/2021

Step 1

Step 1: We are given the series $S = 1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+\ldots \infty$. Show more…

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If $|a|<1$ and $|b|<1$, then the sum of the series $1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+$ $\ldots \infty$ is equal to (A) $\frac{1}{(1-b)(1-a b)}$ (B) $\frac{1}{(1-a)(1-a b)}$ (C) $\frac{1}{(1-a)(1-b)}$ (D) None of these

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Key Concepts

Convergence of Infinite Series

This concept involves understanding when an infinite sum converges to a finite value. In problems like these, knowing that conditions such as |a| < 1 and |b| < 1 guarantee the convergence of the series is crucial, as it allows for manipulation and summation formulas to be applied confidently.

Geometric Series and Sum Formula

A geometric series is a series where each term is a constant multiple of the previous term. The sum formula for a geometric series is fundamental in evaluating and simplifying infinite series, especially when the ratio between terms is less than one in absolute value, ensuring convergence.

Nested Series and Series Manipulation

When dealing with nested or compound series, one frequently encounters sums of partial sums that themselves behave like geometric series. Understanding how to rearrange and combine these series using algebraic manipulation is a key technique that simplifies the evaluation of the overall sum.

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