(D) Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$. Multiplying by $x$,we get $xS = x + 2x^2 + 3x^3 + \dots \infty$. Subtracting the two equations: $S

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(D) Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$. Multiplying by $x$,we get $xS = x + 2x^2 + 3x^3 + \dots \infty$. Subtracting the two equations: $S - xS = 1 + (2x - x) + (3x^2 - 2x^2) + (4x^3 - 3x^3) + \dots \infty$. $S(1 - x) = 1 + x + x^2 + x^3 + \dots \infty$. The right side is an infinite geometric series with first term $a = 1$ and common ratio $r = x$. Since $|x| Therefore,$S(1 - x) = \frac{1}{1 - x}$. $S = \frac{1}{(1 - x)^2}$.

Class 11 Mathematics Sequences and Series nth term of special series, Sum to n terms and Infinite number of terms

Medium

If $|x| < 1$,what is the sum of the series $1 + 2x + 3x^2 + 4x^3 + \dots \infty$?

A
$\frac{1}{1 - x}$
B
$\frac{1}{1 + x}$
C
$\frac{1}{(1 + x)^2}$
D
$\frac{1}{(1 - x)^2}$

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nth term of special series, Sum to n terms and Infinite number of terms

Sum of the series $\frac{2}{3} + \frac{8}{9} + \frac{26}{27} + \frac{80}{81} + \dots$ to $n$ terms is

Medium

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The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+\ldots$ is equal to

DifficultJEE MAIN 2021

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If $S$ is the sum of $n$ terms of the series $1 + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + \dots$,then $S = \dots$

Difficult

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$2^2 + 4^2 + 6^2 + \dots + (2n)^2 = \dots$

Difficult

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In the sequence of sets $(1,2,3), (4,5,6), (7,8,9,10), \ldots$,the sum of elements in the $50^{th}$ set is

DifficultAP EAMCET 2002

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If $|x| < 1$,what is the sum of the series $1 + 2x + 3x^2 + 4x^3 + \dots \infty$?

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