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Find the sum of the series (1^3)/1+(1^3+...

Find the sum of the series `(1^3)/1+(1^3+2^3)/(1+3)+(1^3+2^3+3^3)/(1+3+5)+` up to `n` terms.

Text Solution

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Here
`T_(r)=(1^(3)+2^(3)+3^(3)+..+r^(3))/(1+3+5+….+(2r-1))`
`=([(r(r+1))/2]^(2))/(r/2[1+(2r-1)])`
`=([(r(r+1))/2]^(2))/r^(2)`
`=((r+1)^(2))/4`
`rArr` Sum=`sum_(r=1)^(n)((r+1)^(2))/4`
`=1/4(2^(2)+3^(2)+4^(2)+..+(n+1)^(2))`
`=1/4[(1^(2)+2^(2)+3^(2)+4^(2)+....+(n+1)^(2))-1^(2)]`
`=1/4[((n+1)(n+2)(2n+3))/6-1]`
`=(n(2n^(2)+9n+13))/24`
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