What does the series converge to?

$\displaystyle \sum_{k=0}^\infty \frac{k+1}{k!}$

This was on my final tonight, I don't know (or don't remember) how to figure it out.

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- May 15th 2008, 09:30 PMangel.whiteWhat does the series converge to?
What does the series converge to?

$\displaystyle \sum_{k=0}^\infty \frac{k+1}{k!}$

This was on my final tonight, I don't know (or don't remember) how to figure it out. - May 15th 2008, 09:38 PMMathstud28
- May 15th 2008, 09:44 PMMathstud28
Sorry I did not really explain that well

$\displaystyle \sum_{n=0}^{\infty}\frac{n}{n!}=\sum_{n=0}^{\infty }\frac{1}{(n-1)!}=\frac{D[\sum_{n=0}^{\infty}\frac{1^n}{n!}]}{dx}=e^1$

and $\displaystyle \sum_{n=0}^{\infty}\frac{1}{n!}=\sum_{n=0}^{\infty }\frac{1^n}{n!}=e^1$

so $\displaystyle \sum_{n=0}^{\infty}\frac{n+1}{n!}=\sum_{n=0}^{\inf ty}\frac{n}{n!}+\sum_{n=0}^{\infty}\frac{1}{n!}=e+ e=2e$ - May 15th 2008, 10:37 PMIsomorphism

Alternately

$\displaystyle xe^x = \sum_{k=0}^\infty \frac{x^{k+1}}{k!}$

$\displaystyle (xe^x)' = xe^x + e^x = \left(\sum_{k=0}^\infty \frac{x^{k+1}}{k!}\right)' = \sum_{k=0}^\infty \frac{(k+1)x^{k}}{k!}$

Substituting x=1, we get:

$\displaystyle (xe^x + e^x)\bigg{|}_{x=1} = 2e = \sum_{k=0}^\infty \frac{(k+1)}{k!}$ - May 15th 2008, 10:44 PMMoo
The derivative of a constant is 0... :D

All you have to know is that $\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!}=e^x$

$\displaystyle \sum_{n=0}^\infty \frac{n}{n!}=0+\sum_{n={\color{red}1}}^\infty \frac{n}{n!}=\sum_{n=1}^\infty \frac{1}{(n-1)!}$

I did it because I hate negative factorial, and I'm not sure it's correct to leave this n=0 while we have a (n-1)!

Change : n-1 -> n, so n=1 -> n=0

$\displaystyle =\sum_{n=0}^\infty \frac{1}{n!}=e$

Added to the other term, it yields what the previous posters wrote. - May 16th 2008, 01:16 AMangel.white
Thanks guys, I'm a little annoyed that I forgot that >.< It's one of those things I picked up outside of class and never used again and so I forgot it. If I were more astute, I could have figured that out on the final, but I wasn't :(

- May 16th 2008, 03:17 AMMathstud28
- May 16th 2008, 09:17 AMCaptainBlack
It appears that too many of your posts are composed in the small hours given the high proportion of your posts are less than satisfactory.

It would help the site if you took more time and care over your posts. As it is the low quality of your posts is damaging to the reputation of this site.

RonL - May 16th 2008, 11:56 AMMathstud28