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Find the sum of the series 2/(1xx3)+5/(2...

Find the sum of the series `2/(1xx3)+5/(2xx3)xx2+(10)/(3xx4)xx2^2+(17)/(4xx5)xx2^3+ ton` terms.

Text Solution

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`T_(r)=(r^(2)+1)/(r(r+1))2^(r-1)`
`=(2r^(2)-(r^(2)-1))/(r(r+1))2^(r-1)`
`=((2r)/(r+1)-(r-1)/r)2^(r-1)=(rcdot2^(r ))/(r+1)-((r-1)cdot2^(r-1))/r`
=V( r)-V(r-1), where v(r )=`(rcdot2^(r ))/(r+1)`
Required `sum_(r=1)^(n)T_(r)=sum_(r=1)^(n)(V(r )-V(r-1))`
=V(n)-V(0)
`=n/(n+1)2^(n)-0=n/(n+1)2^(n)`
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