Is it possible to write a sum as an integral to solve it?

I was wondering, for example,

Can:

n=11(3n1)(3n+2)

Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals.

But actually writing an integral form. Like

n=11(3n1)(3n+2)=bag(x) dx

What are some general tricks in finding infinite sum series.

Answer

A General Trick

A General Trick for summing this series is to use Telescoping Series:
n=11(3n1)(3n+2)=13limNNn=1(13n113n+2)=13limN[Nn=113n1Nn=113n+2]=13limN[N1n=013n+2Nn=113n+2]=13limN[1213N+2]=16


An Integral Trick

Since
0entdt=1n
for n>0, we can write
n=11(3n1)(3n+2)=n=1130(e(3n1)te(3n+2)t)dt=130e2te5t1e3tdt=130e2tdt=16

Attribution
Source : Link , Question Author : Amad27 , Answer Author : robjohn

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