The value of $3 - 1 + \frac{1}{3} - \frac{1}{9} +  \ldots $ is equal to

Let $P = 3^{1/3} . 3^{2/9} . 3^{3/27} ...\infty$, then $P^{1/3}$ is equal to

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$ will be

The value of $9^{1/3}\times 9^{1/9} \times 9^{1/27} \times .....\infty$ is

The value of $9^\cfrac{1}{3}.9^\cfrac{1}{9}.9^\cfrac{1}{27}...........$ upto $\infty$, is

If $x=1+a+{ a }^{ 2 }+{ a }^{ 3 }+....$ to $\infty \left( \left| a \right| <1 \right) $ and 
$y=1+b+{ b }^{ 2 }+{ b }^{ 3 }+...$ to $\infty \left( \left| b \right| <1 \right) $ then
$1+ab+{ a }^{ 2 }{ b }^{ 2 }+{ a }^{ 3 }{ b }^{ 3 }+...$ to $\infty =\cfrac { xy }{ x+y-1 } $

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

If $x = 1\, + a + {a^2} + ......\infty $, $y = 1\, + b + {b^2}\,\, + ......\infty $ where $\left| a \right| < 1$ and $\left| b \right| < 1$, then $\left( {1 + ab + {a^2}{b^2} + ........\infty } \right) = ?$

Value of $y = {\left( {0.64} \right)^{{{\log } _{0.25}}\left( {\cfrac{1}{3} + \cfrac{1}{{{3^2}}} + \cfrac{1}{{{3^3}}}....upto   \infty } \right)}}$ is :

If $y=x-x^2+x^3-x^4+....\infty$, then value of x will be?

If the sum of the series $2+\frac {\displaystyle 5}{\displaystyle x}+\frac {\displaystyle 25}{\displaystyle x^2}+\frac {\displaystyle 125}{\displaystyle x^3}+....$ is finite, then-

If $x=1+a+a^2+...\infty$ where $|a| <1 $ and $y=1+b+b^2+...\infty$, where $|b| < 1$, then $1+ab+a^2b^2+...\infty =\dfrac{xy}{x+y-1}$.

${x}^{\cfrac{1}{2}}.{x}^{\cfrac{1}{4}}.{x}^{\cfrac{1}{8}}.{x}^{\cfrac{1}{16}}.....$ to $\infty$

The solution of the equation $(8)^{1+|cos x|+|cos x|^2+|cos x|^3+...)}=4^3$ in the interval $(-\pi, \pi)$ are.

The sum of $7+1+.......$

The series $\dfrac{2x}{x+3}+(\dfrac{2x}{x+3})^{2}+(\dfrac{2x}{x+3})^{3}+........\infty$ will have a definite sum when  

Find the sum of $4,2,1,\cdots$ 

If $y=x^{\dfrac {1}{3}}.x^{\dfrac {1}{9}}.x^{\dfrac {1}{27}}......\infty $, then $y =$

If sum of an infinite geometric series is $\dfrac{4}{3}$ and its Ist term is $\dfrac{3}{4}$, then its common ratio is

The value of x that satisfies the relation 
$x=1-x+{ x }^{ 2 }-{ x }^{ 3 }+{ x }^{ 4 }-{ x }^{ 5 }+........\infty $ 

If $x>0$ and $\displaystyle log _{2}x+log _{2}(\sqrt{x})+log _{2} (\sqrt[4]{x})+log _{2}(\sqrt[8]{x})+...\infty =4 ,$then $x=$

What is the sum of the series $ 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + ....$ equal to ?

The sum of the series formed by the sequence $3, \sqrt{3}, 1....... $ upto infinity is : 

In a Geometric progression with common ratio less than $1$, if $n$ approaches $\infty$ then ${ S } _{ \infty  }$ is

Find the sum of the infinite geometric series $1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+.......$

If $p$ is positive, then the sum to infinity of the series, ${1 \over {1 + p}} - {{1 - p} \over {{{(1 + p)}^2}}} + {{{{(1 - p)}^2}} \over {{{(1 + p)}^3}}} - ......$ is

If $f(x) = x - {x^2} + {x^3} - {x^4} + .............\infty $ where $\left| x \right|\langle 1$ then ${f^{ - 1}}(x) = $

If the sum of an infinitely decreasing G.P. is $3$, and the sum of the squares of its terms is $\dfrac {9}{2}$, then the sum of the cubes of the terms is

Sum of the series ${9^{{1 \over 3}}} \times {9^{{1 \over 9}}} \times {9^{{1 \over {27}}}} \times .......$  is equal to

If the expansion in powers of x of the function $\dfrac{1}{(1 - ax)(1 - bx)} , (a \neq b)$ is $a _0 + a _1x + a _2x^2 + .... \, then \, a _n$ is

If the sum of an infinite $G.P.$ is $1$ and the second term is $'x'$.

The value of $a^{\log _{2}}x$, where $a=0.2,b=\sqrt {5},x=\dfrac {1}{4}+\dfrac {1}{8}+\dfrac {1}{16}+.....$ to $\infty $ is

If $0<x,y,a,b<1$,then the sum of infinite terms of the series $\sqrt x (\sqrt a  + \sqrt x ) + \sqrt x (\sqrt {ab}  + \sqrt {xy} ) + \sqrt x (b\sqrt a  + y\sqrt x ) + .......$ is

If $A = 1 + {r^a} + {r^{2a}} + {r^{3a}}......\infty $ and $B = 1 + {r^b} + {r^{2b}}......\infty$ then$\dfrac{a}{b} = $

The sum of the terms of an infinitely decreasing G.P. is $S$. The sum of the squares of the terms of the progression is -

In a GP the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP up to infinite terms is-

Sum to infinity of a G.P is $15$, whose first term is $a$ then a MUST satisfy the inequality given by

If $x=\sqrt{4}.\sqrt[4]{4}. \sqrt[8]{4}.\sqrt[16]{4}........ \infty$, then 

The sum of  $3,1,\dfrac 13 ,....$ is

If the sum of an infinite GP is 20 and sum of their square is 100 then common ratio will be 

For first $n$ natural numbers we have the following results with usual notations $ \displaystyle \sum _{r=1}^{n}r =\frac{n(n+1)}{2}, \sum _{r=1}^{n}r^{2} =\frac{n(n+1)(2n+1)}{6},\sum _{r=1}^{n}r^{3}=\left ( \sum _{r=1}^{n}r \right )^{2}$ If $\displaystyle a _{1}a _{2}....a _{n} \in A.P $ then sum to $n$ terms of the sequence $\displaystyle \frac{1}{a _{1}a _{2}},\frac{1}{a _{2}a _{3}},...\frac{1}{a _{n-1}a _{n}}$ is equal to $\displaystyle \frac{n-1}{a _{1}a _{n}}$
 and the sum to $ n$ terms of a $G.P$ with first term '$a$' & common ratio '$r$' is given by  $\displaystyle S _{n}= \frac{lr-a}{r-1}$ for $ r \neq 1 $ for $ r =1 $ sum to $n$ terms of same $G.P.$ is $n$ $a$, where the sum to infinite terms of$G.P.$ is the limiting value of
 $\displaystyle \frac{lr-a}{r-1} $ when $\displaystyle n \rightarrow \infty ,\left |  r \right | < l $ where $l$ is the last term of $G.P.$  On the basis of above data answer the following questionsThe sum to infinite terms of the series $\displaystyle \frac{1}{2}+\frac{1}{6}+\frac{1}{18}+.. $ is equal to ?

If $\displaystyle x=\sum _{a=0}^{\infty }a^{n},y=\sum _{a=0}^{\infty }b^{n},z=\sum _{a=0}^{\infty }c^{n}$ Where $a,b,c $ are in A.P and $\displaystyle \left | a \right |<1,\left | b \right |<1,\left | c \right |<1$ then $x,y,z$ are in

If $R \subset\left ( 0,\pi  \right )$ denote the set of values of which satisfies the equation $ \displaystyle 2^{\left ( 1+\left | \cos x \right |+\left | cos^{2}x \right |+\left | cos^{3}x \right | \right )+\left | cos^{4}x  \right |...............\infty}=4$ then $R$ equals

The sum of the series
$\dfrac { 1 } { 1.2 } - \dfrac { 1 } { 2.3 } + \dfrac { 1 } { 3.4 } \ldots \ldots \ldots$  up to  $\infty$  is equal to

The sum of the infinite series, ${ 1 }^{ 2 }-\frac { { 2 }^{ 2 } }{ 5 } +\frac { { 3 }^{ 2 } }{ { 5 }^{ 2 } } -\frac { { 4 }^{ 2 } }{ { 5 }^{ 3 } } +\frac { { 5 }^{ 2 } }{ { 5 }^{ 4 } } -\frac { { 6 }^{ 2 } }{ { 5 }^{ 5 } } +.........$ is :

The first term of an infinitely decreasing G.P. is unity and its sum is S. The sum of the squares of the terms of the progression is

 If  $0<\phi < \pi /2,$   and
 $x= \sum _{n=0}^{\infty} \cos ^{2n} \phi$, $ y=\sum _{n=0}^{\infty } \sin ^{2n} \phi$                     
and $z=\sum _{n=0}^{\infty} \cos ^{2n} \phi \sin ^{2n} \phi $ 
then

Find the sum of the infinite geometric series where the beginning term is $-1$ and the common ratio is $\dfrac{1}{2}$.

$1 + x + x^2 + x^3 +......$ = ?

If $a=\sum _{ n=0 } ^{\infty  }{x^n } ,b=\sum _{n=0  }^{ \infty  }{ y^n } , c=\sum _{n=0  }^{ \infty  }{ (xy)^n } $ where $|x| ,| y| < 1$ ; then

Sum to infinity of the series $\displaystyle \frac { 2 }{ 3 } -\frac { 5 }{ 6 } +\frac { 2 }{ 3 } -\frac { 11 }{ 24 } +...$ is

If $S$ is the sum to infinity of a GP, whose first term is $a$, then the sum of the first $ n$  terms is

$\displaystyle2+1+\frac{1}{2}+\frac{1}{4}+\cdots\cdots\infty$ is

What is the sum of the infinite geometric series where the beginning term is $2$ and the common ratio is $3$?

The value of the infinite product $6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times .........$ is

Calculate the sum of the infinite series: $1 - \dfrac {1}{3} + \dfrac {1}{9} - \dfrac {1}{27} + .....$.

Calculate the sum of the infinite geometric series $2+\left(-\displaystyle\frac{1}{2}\right)+\left(\displaystyle\frac{1}{8}\right)+\left(-\displaystyle\frac{1}{32}\right)+...$

The sum of first $n$ terms of an infinite G.P. is

If ${S} _{p}$ denote the sum of the series $1+{r}^{p}+{r}^{2p}+..$ upto infinity and ${X} _{p}$ be the sum of the series $1-{r}^{p}+{r}^{2p}-..$ upto infinity then $\left( r\in \left( -1,1 \right) -\left{ 0 \right}  \right)$

The sum of an infinite geometric series whose first term is a and common ratio is r is given by

If $S _{1}, S _{2}, S _{3}$ are respectively the sum of n, 2n and 3n terms of a G.P. Then  $S _{1}(S _{3}-S _{2}) = (S _{2} -S _{1})^{2}$.

If $|x| > 1$, then
$\left(1-\dfrac{1}{x}\right)+\left(1-\dfrac{1}{x}\right)^2+\left(1-\dfrac{1}{x}\right)^3+.....=$

If $e^{\displaystyle \left [ \left ( \sin^{2}x + \sin^{4}x + \sin^{6}x + .... + \infty \right ) \log _{e}2\right ]}$ satisfies the equation $\displaystyle x^{2} -9x + 8 = 0$,then the value of $\displaystyle g \left ( x \right ) = \frac{\cos x}{\cos x + \sin x}$ is

lf $e^{(\cos^{2}x+\cos^{4}x+\cos^{6}x+\ldots.)\log 3}$ satisfies $y^{ 2 }-10y+9=0$ and $0\le x\le \cfrac { \pi  }{ 2 } $, then $\cot^{2}x=$

If the sum of an infinite $GP$ is $20$ and sum of their square is $100$ then common ration will be=

For $0 < \phi < \pi/2$ if $x=\sum _{n=0}^{\infty }\cos ^{2n} \phi, y=\sum _{n=0}^{\infty }\sin ^{2n} \phi, z=\sum _{n=0}^{\infty }\cos ^{2n} \phi \sin^{2n}\phi$, then 

The sum of the intercepts cut off by the axes on the lines  $ x+y=a,x+y=ar,x+y=ar^{2}\ldots\ldots\ldots$ where $a\neq 0$ and $r=\displaystyle \dfrac{1}{2}$  is