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I want to evaluate this sum $$\sum_{k=0}^{n}k^{2}2^{k}$$ by summation by parts (two times) and I need to know, if my approach was right.

I know the formula for summation by parts is $$\sum u\Delta v=uv-\sum\left(Ev\right)\Delta u$$ where $E\left(v\left(x\right)\right)=v\left(x+1\right)$ and $(v(x))=v(x+1)-vx)$.

Further if $f$ is a Antiderivative of $g$ I know the formula $$\sum_{k=a}^{b}g\left(k\right)=f\left(b+1\right)-f\left(a\right)$$

First I choose $u=x^{2}$. This means $\Delta u=\Delta x^{2}=2x+1$. Choose $v=2^{x}$. That means $\Delta v=2^{x}$. Then I get

$$\begin{eqnarray*} \sum_{k=0}^{n}k^{2}2^{k} &=& \sum_{0}^{n+1}x^{2}2^{x} \\ &=& \left.x^{2}*2^{x}\right|_{0}^{n+1}-\sum_{0}^{n+1}2^{x+1}(2x+1) \\ &=& (n+1)^2 2^{n+1}-\sum_{0}^{n+1}2x2^{x+1}+2^{x+1} \\ &=& (n+1)^2 2^{n+1}-4\sum_{0}^{n+1}x2^{x} - 2\sum_{0}^{n+1}2^{x} \end{eqnarray*}$$

Now choose $u(x)=x$. This means $\Delta u=\Delta x=1$. Chose $v=2^{x}$. This means $\Delta v=2^{x}$ and $E(v(x))=2^{x+1}$. I get then

$$\begin{eqnarray*} \sum_{x=0}^{n+1}x2^{x+1} &=& \left[x2^{x}-2^{x+1}\right]_{0}^{n+1} \\ &=& ((n+1)-1)2^{(n+1)+1}+2 \\ &=& n2^{n+2}+2 \end{eqnarray*}$$

Putting those two calculations together resolves in $$\begin{eqnarray*} \sum_{k=0}^{n}k^{2}2^{k} &=&(n+1)^2 2^{n+1}-2\sum_{0}^{n+1}x2^{x+1} - \sum_{0}^{n+1}2^{x+1} \\ &=& (n+1)^2 2^{n+1}-2(n2^{n+2}+2) - (n+2)2^{x} \\ &=& 2^{n+1}(n^2+2n+1) -4(n2^{n+2}+2) - 2(n+2)2^{x} \\ &=& 2^{n+1}(n^2+2n+1-8n-(n+2))-8 \\ &=& 2^{n+1}(n^2-7n-1)-8 \\ \end{eqnarray*}$$

But Mathematica says $$2^{n+1} ((n-2) n+3)-6 = 2^{n+1} (n^2-2n+3)-6$$ Where is my fault?

Update

I placed the right calculations as an answer below.

Martin Sleziak
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Aufwind
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  • To format $$\begin{eqnarray} \sum_{k=0}^{n}k^{2}2^{k} &=&2(n+1)2^{n+1}-\left( 2^{n+4}-2^{n+5}\right) =2(n+1)2^{n+1}-2^{n+4}+2^{n+5} \ &=&2^{n+2}+2^{n+3}-2^{n+4}+2^{n+5} \end{eqnarray}$$ you can use blacklash begin{eqnarray} \sum_{k=0}^{n}k^{2}2^{k} &=&2(n+1)2^{n+1}-\left( 2^{n+4}-2^{n+5}\right) =2(n+1)2^{n+1}-2^{n+4}+2^{n+5} \ &=&2^{n+2}+2^{n+3}-2^{n+4}+2^{n+5} \end{eqnarray} – Américo Tavares Apr 30 '11 at 10:39
  • $2(n+1)2^{n+1}\neq 2^{n+2}+2^{n+3}$ – Américo Tavares Apr 30 '11 at 10:42
  • If I may nitpick...you start by saying "I want to solve this equation..." but an equation is something with an equals sign in it, which you don't have. Rather, "I want to evaluate this sum..." – Gerry Myerson Apr 30 '11 at 11:22
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    @Gerry: You are right. I corrected that. If a "nitpick" serves me to learn something new - which I appreciate every time - I am absolutely ok, with that! ;-) – Aufwind Apr 30 '11 at 11:41
  • @Américo: Thanks, I corrected both. – Aufwind Apr 30 '11 at 13:56
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    +1 for trying on your own and showing what you have done. Maybe it will be a lesson to others that you got a specific answer (but I suspect not). – Ross Millikan Apr 30 '11 at 14:51
  • @Auf Your last paragraph is quite wrong. It seems you miscopied something and left an $x$ which then vanished. – Phira Apr 30 '11 at 15:52

2 Answers2

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You wrongly calculated the difference of $x^2$. Plug it directly into the definition.

(And you can always check your calculation by looking at small values of $n$. In particular, you should have 0 for $n=0$. To locate the error, "binary search" is advisable: Put $n=0$ somewhere in the middle to see if the error was in the first half or the second half, etc.)

Phira
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  • I see, thanks. Of course it should be: $\Delta x^2 = (x+1)^{2}-x^{2}=x^{2}+2x+1-x^{2}=2x+1$. I'll edit it right now. – Aufwind Apr 30 '11 at 11:58
7

I calculated it again by pen and paper after letting a little time passing. And I got it! I made some basic mistakes, which I am ashamed of now, that I see them. But you learn by any *kind* of mistakes, don't ya? ;-)

Here the correct calculation:

First I calculated

\begin{equation*} \sum_{k=0}^{n} k 2^{k} \end{equation*}

by summation by parts. I chose $x=u$ and $\Delta 2^x=v$. This gave me $\Delta x=1$ and $v=2^x$. I see the descrete integration of $2^x$ behaves like the analytical integration of $e^x$. (I hope the vocabular I used here is correct, English is not my native language. :-))

$$ \begin{eqnarray*} \sum_{k=0}^{n} k 2^{k} &=& \sum_{0}^{n+1}x2^{x} \\ &=& \left[x2^{x}\right]_{0}^{n+1}-\sum_{0}^{n+1}2^{x+1}\cdot 1 \\ &=& \left(n+1\right)2^{n+1}-2\sum_{0}^{n+1}2^{x} \\ &=& \left(n+1\right)2^{n+1}-2\left(2^{n+1}-1\right) \\ &=& n2^{n+1}+2^{n+1}-2\cdot2^{n+1}+2 \\ &=& n2^{n+1}-2^{n+1}+2 \\ &=& 2^{n+1}\left(n-1\right)+2 \end{eqnarray*} $$ Now I can evaluate

\begin{equation*} \sum_{k=0}^{n} k^2 2^{k} \end{equation*}

For that I chose $x^2=u$ and $\Delta 2^x=v$. This gave me $\Delta x=2x+1$ and $v=2^x$. Now summation by parts and some basic calculations yields

$$ \begin{eqnarray*} \sum_{k=0}^{n}k^{2}2^{k} &=& \sum_{0}^{n+1}x^{2}2^{x} \\

&=& \left[x^{2}2^{x}\right]{0}^{n+1}-\sum{0}^{n+1}2^{x+1}\left(2x+1\right) \ &=& \left(n+1\right)^{2}2^{n+1}-\left(\sum_{0}^{n+1}2x2^{x+1}+2^{x+1}\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4\sum_{0}^{n+1}x2^{x}+2\sum_{0}^{n+1}2^{x}\right)\ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4\left(\left(n-1\right)2^{n+1}+2\right)+2\left[2^{x}\right]_{0}^{n+1}\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4\left(\left(n-1\right)2^{n+1}+2\right)+2\left(2^{n+1}-1\right)\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4\left(\left(n2^{n+1}-2^{n+1}\right)+2\right)+2\cdot2^{n+1}-2\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4\left(n2^{n+1}-2^{n+1}\right)+8+2\cdot2^{n+1}-2\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-\left(4n2^{n+1}-42^{n+1}+8+2\cdot2^{n+1}-2\right) \ &=& \left(n^{2}+2n+1\right)2^{n+1}-4n2^{n+1}+42^{n+1}-8-2\cdot2^{n+1}+2 \ &=& 2^{n+1}\left(\left(n^{2}+2n+1\right)-4n+4-2\right)-8+2 \ &=& 2^{n+1}\left(n^{2}-2n+3\right)-6 \ &=& 2^{n+1}\left((n-2)n+3\right)-6

\end{eqnarray*} $$

which is exacly what WolframAlpha proclaims. Thanks to anyone who suggested corrections. :-)

Aufwind
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