[SOLVED] Find sum of infinite series?

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• Apr 23rd 2009, 06:34 AM
fardeen_gen
[SOLVED] Find sum of infinite series?
Find the sum of the following infinite series:

(n = 0 to ∞) (1/n![(k = 0 to n) ((k+1).∫(from 0 to 1) 2^(-(k +1)x) dx]8)
• Apr 25th 2009, 10:55 PM
fardeen_gen
Is it something to do with binomial or A.P/G.P/H.P? I cannot make head or tail still.
• Apr 26th 2009, 01:40 AM
Moo
Just a question : can you possibly learn latex ? ^^
• Apr 27th 2009, 09:57 PM
fardeen_gen
$\sum\limits_{n = 0}^{ \infty } {\frac{1}{n!}[\sum\limits_{k = 0}^{n} {(k + 1)\left(\int_0^1 2^{-(k + 1)x}\ dx\right)]8}}$

Thanks for the push Moo! :) Can anyone help now?

EDIT: Why is this not working(I want to make the square brackets big):
Code:

\sum\limits_{n = 0}^{ \infty } {\frac{1}{n!}\left[\sum\limits_{k = 0}^{n} {(k + 1)\left(\int_0^1 2^{-(k + 1)x}\ dx\right)\right]8}}
• Apr 28th 2009, 02:58 AM
Moo
Quote:

Originally Posted by fardeen_gen
$\sum\limits_{n = 0}^{ \infty } {\frac{1}{n!}[\sum\limits_{k = 0}^{n} {(k + 1)\left(\int_0^1 2^{-(k + 1)x}\ dx\right)]8}}$

Thanks for the push Moo! :) Can anyone help now?

EDIT: Why is this not working(I want to make the square brackets big):
\sum\limits_{n = 0}^{ \infty } {\frac{1}{n!}\left[\sum\limits_{k = 0}^{n} {(k + 1)\left(\int_0^1 2^{-(k + 1)x}\ dx\right)\right]8}}

$\sum\limits_{n = 0}^{ \infty } \frac{1}{n!}\left[\sum\limits_{k = 0}^{n} (k + 1)\left(\int_0^1 2^{-(k + 1)x}\ dx\right)\right]8$
I removed the red brackets ;)

Anyway, for your problem... :

We have
\begin{aligned}
\int_0^1 2^{-(k+1)x} ~dx &=\int_0^1 \exp\left(-(k+1)x\ln(2)\right) ~dx \\
&=\left. -\frac{1}{(k+1)\ln(2)} \cdot \exp\left(-(k+1)x\ln(2)\right)\right|_0^1 \\
&=\left. -\frac{1}{(k+1)\ln(2)} \cdot 2^{-(k+1)x}\right|_0^1 \\
&=-\frac{1}{(k+1)\ln(2)} \cdot (2^{-(k+1)}-1)
\end{aligned}

$\int_0^1 2^{-(k+1)x} ~dx = \frac{1}{(k+1)\ln(2)} \cdot \left(1-\frac{1}{2^{k+1}}\right)$

So
\begin{aligned} \sum_{k=0}^n (k+1)\int_0^1 2^{-(k+1)x} ~dx
&=\frac{1}{\ln(2)} \sum_{k=0}^n 1-\frac{1}{2^{k+1}} \\
&=\frac{1}{\ln(2)} \left(\sum_{k=0}^n 1 -\sum_{k=0}^n \frac{1}{2^{k+1}}\right) \\
&=\frac{1}{\ln(2)} \left((n+1)-\frac 12 \cdot \frac{1-\frac{1}{2^{n+1}}}{1-\frac 12}\right) \\
&=\frac{1}{\ln(2)} \left((n+1)-\left(1-\frac{1}{2^{n+1}}\right)\right)
\end{aligned}

\begin{aligned} \sum_{k=0}^n (k+1)\int_0^1 2^{-(k+1)x} ~dx
&=\frac{1}{\ln(2)} \left(n+\frac{1}{2^{n+1}}\right) \end{aligned}

Now, we have your initial series equal to :
\begin{aligned}
&=\frac{1}{\ln(2)} \sum_{n=0}^\infty \frac{\left(n+\frac{1}{2^{n+1}}\right)}{n!} \\
&=\frac{1}{\ln(2)} \left( \sum_{n=0}^\infty \frac{n}{n!}+\sum_{n=0}^\infty \frac{(\frac 12)^{n+1}}{n!}\right) \\
&=\frac{1}{\ln(2)} \left( \sum_{n=1}^\infty \frac{n}{n!}+\frac 12 \sum_{n=0}^\infty \frac{(\frac 12)^n}{n!}\right) \end{aligned}

$=\frac{1}{\ln(2)} \left( \sum_{n=1}^\infty \frac{1}{(n-1)!}+\frac 12 \sum_{n=0}^\infty \frac{(\frac 12)^n}{n!}\right)
$

$=\frac{1}{\ln(2)} \left( \sum_{n=0}^\infty \frac{1}{n!}+\frac 12 \sum_{n=0}^\infty \frac{(\frac 12)^n}{n!}\right)$

Now, recall this series : $\sum_{n=0}^\infty \frac{x^n}{n!}=e^x$
For the first one : x=1, for the second one, x=1/2

And it should be finished.

HOwever, I don't quite see the interest of calculating such a series :/
• Apr 28th 2009, 05:50 AM
fardeen_gen
Thanks Moo! (Happy)

The answer: $\frac{2e + \sqrt{2}}{2\ln2}$

Quote:

HOwever, I don't quite see the interest of calculating such a series :/
I don't see the interest either. But we got to do what we got to do :)

EDIT: My typo :D
Answer: $\frac{2e + \sqrt{e}}{2\ln2}$
• Apr 28th 2009, 10:33 AM
Moo
Quote:

Originally Posted by fardeen_gen
Thanks Moo! (Happy)

The answer: $\frac{2e + \sqrt{2}}{2\ln2}$

I don't see the interest either. But we got to do what we got to do :)

Hmm I think it's rather $\frac{2e+\sqrt{{\color{red}e}}}{2 \ln(2)}$, isn't it ? (Surprised)

You're welcome. And you're right ^^