Can be series \sum_{r=0}^\infty\frac{e^{-r}}{1+r} be further simplified?

Guy Rollins

Guy Rollins

Answered question

2022-02-28

Can be series r=0er1+r be further simplified?

Answer & Explanation

Balraj Conrad

Balraj Conrad

Beginner2022-03-01Added 9 answers

Start with the series
11t=n=0tn,
and integrate term-by-term from t=0 to t=x to get
ln(1x)=n=0xn+1n+1
Then, divide by x to get
ln(1x)x=n=0xnn+1
Finally, plug in x=e1 to get
n=0enn+1=eln(1e1)

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