Question

`1/2(1/2+1/3)-1/4(1)/(2^(2))+(1)/(3^(2))+1/6(1)/(2^(3))+(1)/(3^(3))+…infty` is equal to
A. `log_(e)2`
B. `log_(e)3`
C. `log_(e)sqrt(2)`
D. `log_(e)sqrt(3)`

Answer 1

Answer:
We have
`1/2(1/2+1/3)-1/4(1)/(2)^(2)+(1)/(3^(2))+1/6(1)/(2^(3))+(1)/(3^(3))+..infty`
`=(1)/(2.1)-1/4.(1)/(2^(2))+1/6.(1)/(2^(3))..infty`
`=1/2log_(e)(1+1/2)+1/2log_(e)(1+1/3)`
`=1/2{log_(e)(3/2)+log_(e)(4/3)}=1/2log_(e)2=log_(e)sqrt(2)`