# Thread: Find all real numbers such that the series converges!

1. ## Find all real numbers such that the series converges!

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.

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3. ## Re: Find all real numbers such that the series converges!

Yes I knew that and I did come to the harmonic series conclusion, but isn't it true that lim as n→∞ (1+1/n)^n is different from summation (1+1/n)^n as n from 1 to infinity?

4. ## Re: Find all real numbers such that the series converges!

yeah ok, let me think about it some.

Sure thanks!

6. ## Re: Find all real numbers such that the series converges!

Also since you are answering all my questions, I might as well post one more for you to look at. That doesn't look too hard, but there's got to be some trick that I can't think of at the moment!

7. ## Re: Find all real numbers such that the series converges!

Originally Posted by mynameisanthonyg2013
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?
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 $\sum\dfrac{c}{n}$.

8. ## Re: Find all real numbers such that the series converges!

So use the compassion test with $\displaystyle \sum \frac{c}{n}$.
do you mean comparison test?