If |x|<1, then the coefficient of x^n in expansion of (1 .+x+x^2+x^3+…)^2 is: (A) n (B) n-1 (C)

Name: Problem 122, Chapter 9: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-09-20T02:34:21-08:00
Duration: 1 min 7 s
Channel: Aman Gupta
Description: If |x|<1, then the coefficient of x^n in expansion of (1 .+x+x^2+x^3+…)^2 is: (A) n (B) n-1 (C) n+2 (D) n+1

step 1 in this problem since mod x is less than 1 we can write 1 plus x squared plus x cube whole squar

Question

If $|x|<1$, then the coefficient of $x^{n}$ in expansion of $(1$ $\left.+x+x^{2}+x^{3}+\ldots\right)^{2}$ is: (A) $n$ (B) $n-1$ (C) $n+2$ (D) $n+1$

   If $|x|<1$, then the coefficient of $x^{n}$ in expansion of $(1$ $\left.+x+x^{2}+x^{3}+\ldots\right)^{2}$ is:
(A) $n$
(B) $n-1$
(C) $n+2$
(D) $n+1$

A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar 2019 Edition

Chapter 9, Problem 122

Instant Answer

Solved by Expert Aman Gupta

09/20/2021

Step 1

Since $|x|<1$, we can use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$ where $a$ is the first term and $r$ is the common ratio. In this case, $a=1$ and $r=x$, so the sum of the series is $\frac{1}{1-x}$. Show more…

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If $|x|<1$, then the coefficient of $x^{n}$ in expansion of $(1$ $\left.+x+x^{2}+x^{3}+\ldots\right)^{2}$ is: (A) $n$ (B) $n-1$ (C) $n+2$ (D) $n+1$

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Key Concepts

Geometric Series

A geometric series is a series of the form 1 + x + x^2 + x^3 + …, which converges for |x| < 1. This series can be summed using the formula 1/(1 - x), and is foundational for understanding series expansion and power series in calculus and analysis.

Generating Functions

Generating functions encode sequences as coefficients in a power series. In this problem, the series (1 + x + x^2 + …) is used as a generating function for the constant sequence 1, and when squared, it generates a new sequence whose nth term corresponds to the number of ways two indices can sum to n.

Series Convolution (Cauchy Product)

When two power series are multiplied together, the coefficient of x^n in the product is obtained by summing the products of coefficients whose indices add up to n. This process, known as the Cauchy product or convolution of series, explains why squaring the geometric series yields coefficients that increment linearly with n.

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