closed-form term for this sum:

Question

Is there an easy closed-form term for

$$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$

thus when the sum starts at a constant $k$ instead of $1$?

EDIT: Thanks for your help. Is there a Chance to solve this sum-term? Because this is not really what I expect, when I talk about a closed-form term.

A Little bit more of context might help, maybe:

I have $$f(n,p)=\sum_{j=k}^{\infty} \frac{(np)^j}{j!} e^{-np}$$ and it is meant that the partial Derivation is $$\frac{\delta f(n,p)}{\delta n}=\frac{p (np)^{k-1}}{(k-1)!}e^{-np}$$ but I have no idea how to get to this.

Because to me:

$$\frac{\delta f(n,p)}{\delta n}=\sum_{j=k}^{\infty} \left( \frac{p (np)^{j-1}}{j!} e^{-np} -\frac{p (np)^j}{j!} e^{-np} \right)$$ but then I am stuck.

From $\displaystyle{\large\tt M}$: $\displaystyle{\large\frac{\Gamma (k)-\Gamma (k,x)}{\Gamma (k)}}$. — Felix Marin, Jul 07 '14 at 17:03
@FelixMarin Could you specify what you mean with that? I know about the Gamma and the incomplete Gamma functions, but I can't see how to apply this here. — user136457, Jul 07 '14 at 17:05

score 1 · Answer 1 · edited Jul 07 '14 at 13:41

1

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

edited Jul 07 '14 at 13:41

Jam

10,325

answered Jul 07 '14 at 12:47

draks ...

18,449

thanks for your answer. I stated above in the question the whole Problem, pointing out what I could mean with a closed form, because somehow I don't get how they got to this solution. – user136457 Jul 07 '14 at 12:49
@mookid correct but my sum's upper limit is $k-1$, or did I miss something... – draks ... Jul 07 '14 at 14:34
you are right. But I don't know if it is what OP was looking for.... – mookid Jul 07 '14 at 14:44

score 0 · Answer 2 · answered Jul 07 '14 at 12:43

0

Yes:

$$e^{-x}\sum_{j=k}^\infty \frac 1{j!} x^j = e^{-x}\left[\exp x - \sum_{j=0}^{k-1} \frac 1{j!} x^j \right] =e^{-x}\int_0^x \frac{(x-t)^{k-1}}{(k-1)!} e^{t} dt $$

answered Jul 07 '14 at 12:43

mookid

28,236

1

$x$ as limit and inside the integral looks a little strange. Are you sure? – draks ... Jul 07 '14 at 12:50
x does not appear as a limit, does it? – mookid Jul 07 '14 at 13:02
$e^{-x}\int_0^{\color{red}x}$... – draks ... Jul 07 '14 at 14:32
there is no error, see http://en.wikipedia.org/wiki/Taylor's_theorem#Derivation_for_the_integral_form_of_the_remainder – mookid Jul 07 '14 at 14:43
@draks...: it would only seem strange to me if $x$ were the variable of integration, but $t$ is the variable of integration. – robjohn Jul 08 '14 at 15:51

score 0 · Accepted Answer · answered Jul 08 '14 at 07:56

You just made a mistake when differentiating $$f(x)=\sum_{j=k}^{\infty} \frac{(xp)^j}{j!} e^{-xp}.$$ The actual derivative is $$f'(x)=\sum_{j=k}^{\infty} \left( \color{red}{j}\,\frac{p\,(xp)^{j-1}}{j!} e^{-xp} -\frac{p\,(xp)^j}{j!} e^{-xp} \right),$$ that is, provided $k\geqslant1$, using the change of indices $\ell=j-1$ in the first summation, $$f'(x)=\sum_{\ell=k-1}^{\infty}\frac{p(xp)^{\ell}}{\ell!} e^{-xp} -\sum_{j=k}^{\infty} \frac{p (xp)^j}{j!} e^{-xp}=\frac{p(xp)^{k-1}}{(k-1)!} e^{-xp}.$$

score 0 · Answer 4 · answered Jul 08 '14 at 08:43

0

Incomplete Gamma function and Eq.2 in : http://mathworld.wolfram.com/IncompleteGammaFunction.html

enter image description here

answered Jul 08 '14 at 08:43

JJacquelin

66,221
3
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closed-form term for this sum:

4 Answers4