Test the condition for convergence of \sum_{n=1}^\infty\frac{1}{n(n+1)(n+2)} and find the sum if

Sandra Allison 2022-01-18 Answered
Test the condition for convergence of
n=11n(n+1)(n+2)
and find the sum if it exists.
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David Clayton
Answered 2022-01-18 Author has 36 answers
Karen Robbins
Answered 2022-01-19 Author has 49 answers

There is an alternate method and is as follows.
Notice that
1n(n+1)(n+2)=(n1)!(n+2)!=12B(n,3)
where B(x,y) is the Beta function. Using an integral form of the Beta function the summation becomes
S=n=11n(n+1)(n+2}
=1201(n=1xn1)(1x)2dx
=1201(1x)21xdx=1201(1x)dx
=14
This leads to the known result
n=11n(n+1)(n+2)=14

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alenahelenash
Answered 2022-01-24 Author has 343 answers
Alternatively, take 11x=n=1xn1 and integrate three times with lower limit 0, giving log(1x)=n=1xnn x+(1x)log(1x)=n=1xnn(n+1) 34x212x12(1x)2log(1x)=n=1xnn(n+1)(n+2)
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