(D) The sum of an infinite $G.P.$ is given by the formula $S_{\infty} = \frac{a}{1 - r}$.This formula is valid only when the common ratio $r$ satisfie

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Only for $-1 < r < 1$ $(r \ne 0)$

(D) The sum of an infinite $G.P.$ is given by the formula $S_{\infty} = \frac{a}{1 - r}$.This formula is valid only when the common ratio $r$ satisfies the condition $|r| < 1$,which means $-1 < r < 1$.

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

Medium

The sum of an infinite $G.P.$ with common ratio $r$ can be found:

A
For all values of $r$
B
For only positive values of $r$
C
Only for $0 < r < 1$
D
Only for $-1 < r < 1$ $(r \ne 0)$

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Class 11

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Sequences and Series

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

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=a_{1}=0$ and $a_{n+2}=2a_{n+1}-a_{n}+1$ for all $n \geq 0$. Then,$\sum\limits_{n=2}^{\infty} \frac{a_{n}}{7^{n}}$ is equal to

DifficultJEE MAIN 2022

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The sum to $n$ terms of $(2n - 1) + 2(2n - 3) + 3(2n - 5) + \dots$ is

Medium

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Show that $\frac{1 \times 2^{2}+2 \times 3^{2}+\ldots+n \times(n+1)^{2}}{1^{2} \times 2+2^{2} \times 3+\ldots+n^{2} \times(n+1)}=\frac{3 n+5}{3 n+1}$

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The product $(32)(32)^{1/6}(32)^{1/36} \dots \infty$ is

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$\sum_{k=1}^5 \frac{1^3+2^3+\ldots+k^3}{1+3+5+\ldots+(2 k-1)}$ is equal to (in $.5$)

DifficultAP EAMCET 2004

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The sum of an infinite $G.P.$ with common ratio $r$ can be found:

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Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=a_{1}=0$ and $a_{n+2}=2a_{n+1}-a_{n}+1$ for all $n \geq 0$. Then,$\sum\limits_{n=2}^{\infty} \frac{a_{n}}{7^{n}}$ is equal to

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