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Find the sum of the infinte series (1)/(...

Find the sum of the infinte series `(1)/(9)+(1)/(18)+(1)/(30)+(1)/(45)+(1)/(63)+…`

A

`(1)/(3)`

B

`(1)/(4)`

C

`(1)/(5)`

D

`(2)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `T_(n)=(1)/(3)[(1)/(3)+(1)/(6)+(1)/(10)+(1)/(15)+(1)/(21)+…]`
`=(2)/(3)[(1)/(2*3)+(1)/(3*4)+(1)/(4*5)+(1)/(5*6)+…]`
Hence `T_(n)=(2)/(3)(1)/((n+1)(n+2))`
`=(2)/(3)[(1)/(n+1)-(1)/(n+2)]`
`:.S_(oo)=(2)/(3)*(1)/(2)=(1)/(3)`
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