(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

The sum of $n$ terms of the series $12 + 16 + 24 + 40 + \dots$ is

Medium

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If the $n^{th}$ term of a sequence is $n(n + 1)$,then what is the sum of its $n$ terms?

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The odd numbers are divided as follows:
Row $1$: $1, 3$
Row $2$: $5, 7, 9, 11$
Row $3$: $13, 15, 17, 19, 21, 23$
Then the sum of the $n^{th}$ row is:

Medium

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Find the sum to $n$ terms of the series $3 \times 1^{2} + 5 \times 2^{2} + 7 \times 3^{2} + \dots$

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If $4^3+8^3+12^3+\ldots$ up to $n$ terms $= k n^2(n+1)^2$ (for all $n \in N$),then $k=$

DifficultAP EAMCET 2017

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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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The odd numbers are divided as follows:
Row $1$: $1, 3$
Row $2$: $5, 7, 9, 11$
Row $3$: $13, 15, 17, 19, 21, 23$
Then the sum of the $n^{th}$ row is: