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lim(nrarroo) Sigma(r=1)^(n) (r)/(1xx3xx5...

`lim_(nrarroo) Sigma_(r=1)^(n) (r)/(1xx3xx5xx7xx9xx...xx(2r+1))` is equal to

A

`1/3`

B

`3/2`

C

`1/2`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`T(r )=r/(1xx3xx5xx…xx(2r+1))`
`=((2r+1-1)/(2(1xx3xx5..(2r+1))))`
`=1/2(1/(1xx3xx5..(2r-1))-1/(1xx3xx5..(2r+1)))`
`=-1/2[V(r )-V(r-1)]`
`rArrsum_(r=1)^(n)T(r )=-1/2(V(n)-V(0))`
`=1/2(1-1/(1xx3xx5xx…xx(2n+1)))`
`rArrlim_(ntooo)underset(r=1)overset(n)Sigmar/(1xx3xx5xx7xx9xx..xx(2r+1))`
`=lim_(ntooo)1/2(1-1/(1xx3xx5xx....xx(2n+1)))=1/2`
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