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Is there a real number c such that the series:
∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent?
I tried to use ratio test but it turned out really messy. Any other ideas?
Thanks.
In the case that $c>0 $ then:
It is well known that ${\left( {1 + {n^{ - 1}}} \right)^n}$ is a monotone increasing sequence $\to e$.
Therefore $(\forall n)\left[e-{\left( {1 + {n^{ - 1}}} \right)^n}>0\right]$. So use the comparison test with $\sum\dfrac{c}{n}$.