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Find the sum Sigma(r=1)^(n) r/((r+1)!). ...

Find the sum `Sigma_(r=1)^(n) r/((r+1)!)`. Also, find the sum of infinite terms.

Text Solution

Verified by Experts

Here, `T_(r)=r/((r+1)!)=((r+1)-1)/((r+1)!)=1/(r!)-1/((r+1)!)`
`=V(r) -V(r+1)`, where `V(r )=1/(r!)`
`thereforesum_(r=1)^(n)T_(r)=sum_(r=1)^(n)(V®-V(r+1))`
`=V(1)-V(n+1)`
`=1-1/((n+1)!)`
`thereforesum_(r=1)^(oo)T_(r)=lim_(ntooo)(1-1/((n+1)!))=1-0=1`
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