Arithmetico-geometric Progression

The sum of infinite series $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\dots$ to $\infty$

Evaluate $1\mathrm{ }+\frac{4}{5}\mathrm{ }+\frac{7}{25}\mathrm{ }+\frac{10}{125}+\dots$ to infinite terms.

Find the value up to infinity $\mathrm{ }{2}^{\frac{1}{4}} .4^{\frac{1}{8}} .\mathrm{ }{8}^{\frac{1}{16}}\dots \mathrm{\infty }$

If ${10}^9+2\left(11\right){\left(10\right)}^8+3.{\left(11\right)}^2{\left(10\right)}^7+\dots +10{\left(11\right)}^9=\mathrm{k}{\left(10\right)}^9,\mathrm{ }$ then value of $\mathrm{ }\mathrm{ }\mathrm{k}\mathrm{ }$ is equal to

The positive integer $n$ for which $2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n=2^{n+10}$ is

$2^{\frac{1}{4}}.\mathrm{ }{4}^{\frac{1}{8}}.\mathrm{ }{8}^{\frac{1}{16}}\dots ..\mathrm{\infty }$ is equal to 

For which positive integer $\mathrm{n}$, is the ratio $\frac{\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{k}}^2}{\sum _{\mathrm{k}=1}^{\mathrm{n}}\mathrm{k}}$ an integer -

The value of sum $\sum _{\mathrm{k}=1}^{\infty }\sum _{\mathrm{n}=1}^{\infty }\frac{\mathrm{k}}{2^{\mathrm{n}+\mathrm{k}}}$ is -

The sum of the infinite series $1\mathrm{ }+\mathrm{ }\left(1+\frac{1}{2}\right)\mathrm{ }\left(\frac{1}{3}\right)+\left(1+\frac{1}{2}+\frac{1}{2^2}\right)\mathrm{ }\left(\frac{1}{3^2}\right)+\mathrm{ }\dots \dots \mathrm{\infty }$ is

If $\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\left[2\mathrm{x}+4{\mathrm{x}}^3+\dots \dots \dots .+2\mathrm{n}{\mathrm{x}}^{2\mathrm{n}–1}\right]\mathrm{ },\mathrm{ }\left(0\mathrm{ }<\mathrm{ }\mathrm{x}\mathrm{ }<\mathrm{ }1\right)$ then $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{d}\mathrm{x}$ is equal to -

Sum to infinity of the series $1-3x+5x^2-7x^3+....$ when $\left|x\right|<1$

$1+\frac{1}{3.2^2}+\frac{1}{5.2^4}+\frac{1}{7.2^6}+....=$

$x\mathrm{l}\mathrm{o}{\mathrm{g}}_{e}a+\frac{x^3}{3!}{\left(\mathrm{l}\mathrm{o}{\mathrm{g}}_{e}a\right)}^3+\frac{x^5}{5!}{\left(\mathrm{l}\mathrm{o}{\mathrm{g}}_{e}a\right)}^5+...=$

If ${\left(10\right)}^9+2{\left(11\right)}^1{\left(10\right)}^8+3{\left(11\right)}^2{\left(10\right)}^7+...+10{\left(11\right)}^9=k{\left(10\right)}^9$ , then $k$ is equal to

Sum to infinity of the series $1-3x+5x^2-7x^3+....$ when $\left|x\right|<1$

If $21(x^2+y^2+z^2)=(x+2y+4z{)}^2$ , then  $x, y, z$ are in

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+ \dots$ is

${99}^{th}$ term of the series $2+7+14+23+34\dots$ is

If $\left|a\right|<1$, then $1+2a+3a^2+4a^3+...$ is equal to

If $\left|x\right|<1$, then the sum of the series $1+2x+3x^2+4x^3+...\infty$ will be