Sum = Integral

**BrownianMan** · September 13th 2011, 01:02 PM

I have to prove that $\sum_{x=0}^{k-1}\frac{(\lambda w)^{x}e^{- \lambda w}}{x!}=\int_{\lambda w}^{\infty }\frac{z^{k-1}e^{-z}}{(k-1)!}dz$

How would I go about doing that?

**CaptainBlack** · September 13th 2011, 02:29 PM

Originally Posted by BrownianMan

I have to prove that $\sum_{x=0}^{k-1}\frac{(\lambda w)^{x}e^{- \lambda w}}{x!}=\int_{\lambda w}^{\infty }\frac{z^{k-1}e^{-z}}{(k-1)!}dz$

How would I go about doing that?

Induction on k

CB

**BrownianMan** · September 13th 2011, 03:59 PM

How do I start?

**CaptainBlack** · September 13th 2011, 07:42 PM

Originally Posted by BrownianMan

How do I start?

You prove a base case, for this show that the formula is true when $k=1$ that is show that:

$\frac{(\lambda w)^0e^{-\lambda w}}{0!}=e^{-\lambda w}=\int_{\lambda w}^{\infty} \frac{z^0 e^{-z}}{0!}\; dz=\int_{\lambda w}^{\infty} e^{-z}\; dz$

That is show:

$e^{-\lambda w}=\int_{\lambda w}^{\infty} e^{-z}\; dz$ .

Next assume it true for some $k$ , then prove it true for $k+1$ . This will involve integration by parts, start by using the induction assumption:

$\sum_{x=0}^{(k+1)-1}\frac{(\lambda w)^xe^{-\lambda w}}{x!}=\sum_{x=0}^{k-1}\frac{(\lambda w)^xe^{-\lambda w}}{x!}+\frac{(\lambda w)^ke^{-\lambda w}}{k!}\\ \\ \\ \phantom{xxxxxxxxx} =\int_{\lambda w}^{\infty} \frac{z^{k-1} e^{-z}}{(k-1)!}\; dz+\frac{(\lambda w)^ke^{-\lambda w}}{k!}$

Now show using integration by parts on the integral on the right that:

$\int_{\lambda w}^{\infty} \frac{z^{k-1} e^{-z}}{(k-1)!}\; dz+\frac{(\lambda w)^ke^{-\lambda w}}{k!}=\int_{\lambda w}^{\infty} \frac{z^{(k+1)-1} e^{-z}}{((k+1)-1)!}\; dz$

CB

**BrownianMan** · September 14th 2011, 09:40 AM

Thanks so much!

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