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Consider a series 1/2+1/(2^2)+2/(2^3)+3/...

Consider a series `1/2+1/(2^2)+2/(2^3)+3/(2^4)+5/(2^5)+.............+(lambdan)/(2^n).` If `S_n` denotes its sum to `n` tems, then `S_n` cannot be

A

2

B

3

C

4

D

5

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`:. S_(n)=(1)/(2)+(1)/(2^(2))+(2)/(2^(3))+(3)/(2^(4))+(5)/(2^(5))+"...."+(lambdan)/(2^(n))`
` =(3)/(4)+(1)/(4)((1)/(2)+(1)/(2^(2))+(2)/(2^(3))+(3)/(2^(4))+(5)/(2^(5))+"...."+(lambdan)/(2^(n))) +(1)/(2)((1)/(2)+(1)/(2^(2))+(2)/(2^(3))+"...."+(lambdan)/(2^(n))) -(1)/(4)-(lambda_(n))/(2^(n+2))-(lambda_(n))/(2^(n+1))`
`implies S_(n)=(3)/(4)+(1)/(4)S_(n)+(1)/(2)S_(n)-(1)/(4)-(lambda_(n))/(2^(n+2))-(lambda_(n))/(2^(n+1))`
`implies =(1)/(4)S_(n)=(1)/(2)-(lambda_(n))/(2^(n+2))-(lambda_(n))/(2^(n+1))implies S_(n)=2-(lambda_(n))/(2^(n+2))-(lambda_(n))/(2^(n-1))lt2`
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