Answer to Question #107201 in Real Analysis for Aarzoo Rawat

Question #107201
Prove that if x is not a negative integer , summation n=1 to infinity of 1/(x+n)(x+n+1) = 1/1+x .
1
Expert's answer
2020-04-01T10:27:51-0400

"\\sum^{\\infty}_{n=1}\\dfrac1{(x+n)(x+n+1)}=\\sum^{\\infty}_{n=1}(\\dfrac1{x+n}-\\dfrac1{x+n+1})="

"=\\lim_{N\\rightarrow \\infty}\\sum^{N}_{n=1}(\\dfrac1{x+n}-\\dfrac1{x+n+1})=\n\\lim_{N\\rightarrow \\infty}(\\dfrac1{x+1}-\\dfrac1{x+N+1})="

"=\\dfrac1{x+1}-\\lim_{N\\rightarrow \\infty}\\dfrac1{x+N+1}=\\dfrac1{x+1}"


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