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Find the sum 1+1/(1+2)+1/(1+2+3)++1/(1+2...

Find the sum `1+1/(1+2)+1/(1+2+3)++1/(1+2+3++n)dot`

Text Solution

Verified by Experts

`T_(r)=1/(1+2+3+..+r)`
`=2/(r(r+1))`
`=2((r+1)-r)/(r(r+1))`
`=2(1/r-1/(r+1))`
`=2(V(r)-V(r+1))`, where V(r )=`1/r`
`thereforesum_(r=1)^(n)T_(r)=2sum__(r=1)^(n)(V(r )-V(r+1))`
`=2(V(1)-V(n+1))`
`2(1-1/(n+1))`
`(2n)/(n+1)`
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