Home
Class 12
MATHS
Evaluate S=sum(n=0)^n(2^n)/((a^(2^n)+1)...

Evaluate `S=sum_(n=0)^n(2^n)/((a^(2^n)+1)` (where `a>1)`.

Text Solution

Verified by Experts

`S=sum_(n=0)^(oo)(2^(n))/(a^(2n)+1)(agt1)`
`S_(n)=sum_(n=0)^(n)(2^(n))/(a^(2n)+1)`
`=(1)/(a+1)+(2)/(a^(2)+1)+(4)/(a^(4)+1)+(8)/(a^(8)+1)+"........"+(2^(n))/(a^(2^n)+1)`
`=(1)/(1+a)+(2)/(1+a^(2))+(4)/(1+a^(4))+(8)/(1+a^(8))+"........"+(2^(n))/(1+a^(2^n))`
`=(-(1)/(1-a)+(1)/(1-a))(1)/(1+a)+(2)/(1+a^(2))+(4)/(1+a^(4))+(8)/(1+a^(8))+"........"+(2^(n))/(1+a^(2^n))`
`=(1)/(a-1)+((1)/(1-a)+(1)/(1+a))+(2)/(1+a^(2))+(4)/(1+a^(4))+"........"+(2^(n))/(1+a^(2^n))`
`=(1)/(a-1)+((2)/(1-a^(2))+(2)/(1+a^(2)))+(4)/(1+a^(4))+"........"+(2^(n))/(1+a^(2^n))`
`" " vdots " "vdots " " vdots " "`
`S_(n)=(1)/(a-1)+(2^(n+1))/(1-a^(2^n+1))`
`S=lim_(n to oo)S_(n)=lim_(n to oo)((1)/(a-1)+(2^(n+1))/(1-a^(2^n+1)))`
`=lim_(n to oo)((1)/(a-1)+((2^(n+1))/(a^(2^n+1)))/((1)/(a^(2^n+1))-1))=(1)/(a-1)+(0)/(0-1)=(1)/(a-1)`.
Promotional Banner

Similar Questions

Explore conceptually related problems

The value of sum_(n=0)^(100)i^(n!) equals (where i=sqrt(-1))

The value of the sum sum_(n=1) ^(13) (i^(n)+i^(n+1)) , where i = sqrt( - 1) ,equals :

The value of the sume sum_(n=1)^(13) ( i^(n) + i^(n+1)) , where i = sqrt( -1) , equals :

The value of sum_(k=1)^(13)(i^(n)+i^(n+1)) , where i=sqrt(-1) equals :

If x^(2)-x+1=0 then the value of sum_(n=1)^(5)(x^(n)+(1)/(x^(n)))^(2) is

If x = sum_(n=0)^(oo) a^(n) .,y=sum_(n=0)^(oo) b^(n) , z = sum_(n=0)^(oo) (ab)^(n) , where a,blt 1 , then :

age0 , then sum_(n=1)^(infty)((a)/(a+1))^(n) equals

lim_(n rarr oo) sum_(r=0)^(n-1) 1/(n+r) =

The nth term of a series is given by t_(n)=(n^(5)+n^(3))/(n^(4)+n^(2)+1) and if sum of its n terms can be expressed as S_(n)=a_(n)^(2)+a+(1)/(b_(n)^(2)+b) where a_(n) and b_(n) are the nth terms of some arithmetic progressions and a, b are some constants, prove that (b_(n))/(a_(n)) is a costant.

lim_(n rarr oo) n.sum_(r=0)^(n-1) 1/(n^(2)+r^(2)) =