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The answer I got by hand is not the same to the one I found using a spreadsheet.

$\displaystyle S = 1 + \frac{3}{4} + \frac{7}{16} + \frac{15}{64} + \frac{31}{256} + \ldots$

$\displaystyle \frac{1}{4}S = \hspace{8.5pt} \frac{1}{4} + \frac{3}{16} + \frac{7}{64} + \frac{15}{256} + \ldots$

$\displaystyle \frac{3}{4}S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \qquad \leftarrow S- \frac{1}{4}S$

For the Infinite Sum on the RHS $\displaystyle \left(S = \frac{a}{1-r}\right)$:

$\displaystyle a = 1$

$\displaystyle r = \frac{1}{2}$

Then

$\displaystyle \frac{3}{4}S = \frac{1}{1-\frac{1}{2}}$

$\displaystyle \frac{3}{4}S = 2$

$\displaystyle S = \frac{8}{3}$

Using Excel the answer is $\displaystyle \frac{5}{3}$, but I don't know where is the issue.

Thanks!!

asd
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    try to learn some python, is better than excel! – L F May 13 '15 at 21:01
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    Why is $\frac34 S $ is what you wrote up there,? – Atvin May 13 '15 at 21:01
  • @Atvin I subtracted the series $\frac{1}{4}S$ from $S$. Thus $S - \frac{1}{4}S = \frac{3}{4}S$. – asd May 13 '15 at 21:03
  • @LuisFelipeVillavicencioLopez for this problem, both will work same and hence suggestion is not useful. – Sufyan Naeem May 13 '15 at 21:03
  • I am not sure that $\frac34 S$ will be the complete $1+\frac12+...$ series, I don't find $\frac14$ – Atvin May 13 '15 at 21:05
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    It is highly likely that you forgot the first term $1$ in Excel. –  May 13 '15 at 21:11
  • I'm voting to close this question as off-topic because: it is about a problem with an Excel model of a mathematical problem. Debugging Excel spreadsheets is off-topic for MSR. – Rob Arthan May 13 '15 at 21:17
  • The terms are all positive and the sum of the first two is clearly greater than $\frac 53$. Basic reality checks always come in handy. I think @YvesDaoust is very likely right. – Mark Bennet May 13 '15 at 21:18
  • Excel can't do infinite summation. Debugging your representation of an infinite series in Excel is not in the scope of MSE. You should be able to calculate the first few partial sums by hand to resolve the inconsistency. – Rob Arthan May 13 '15 at 21:19
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    @RobArthan Take it easy. I just did the thing up to 50 terms (in Excel is a good approximation to infinity). As Yves Daoust says, I forgot to add the first term: $1$. – asd May 13 '15 at 21:22
  • In general, I would strongly advise against using ANY language to evaluate infinite series except for WolframAlpha, Mathematica, Matlab, etc which have support for it. The reason is because simply adding the terms can result in huge errors if convergence is slow. To see a good example of how things can go wrong, try 'evaluate' the harmonic series. – ktoi May 13 '15 at 21:29

5 Answers5

7

Hint: Notice that $$ S = \sum_{i=0}^\infty \frac{2^{i+1}-1}{2^{2i}} $$ and distribute.

William Stagner
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We can pattern match this to a difference of geometric series. The sum in question is $$\begin{align} 1 + \frac{3}{4} + \frac{7}{16} + \frac{15}{64} + \frac{31}{256} = 1+\sum_{k=1}^\infty \frac{2^{k+1}-1}{4^{k}} \\ = 1+\sum_{k=1}^\infty \frac{2^{k+1}}{4^{k}}-\sum_{k=1}^\infty \frac{1}{4^{k}}\end{align}$$ Now use geometric series...

k1.M
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graydad
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Your answer seems to be correct. We see that Excel's answer is wrong by adding just the first 2 terms in the series, $1, \frac 34$ and we see that $1.75>1.66..$, so we can see that Excel's answer is wrong.

Teoc
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At first, note that $S=\dfrac{2^1-1}{4^{1-1}} + \dfrac{2^2-1}{4^{2-1}}+ \dfrac{2^3-1}{4^{3-1}}+\dfrac{2^4-1}{4^{4-1}} + \dfrac{2^5-1}{4^{5-1}}+\ldots $

So: $$S=\displaystyle\sum_{k=1}^{\infty}\dfrac{2^k-1}{4^{k-1}}?$$

$$S=\displaystyle\sum_{k=1}^{\infty}\dfrac{2^k}{4^{k-1}}-\dfrac{1}{4^{k-1}}$$ $$S=\displaystyle\sum_{k=1}^{\infty}\dfrac{4}{2^k}-\dfrac{1}{4^{k-1}}$$ And you can go on since here.

Community
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L F
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When you write "Using Excel the answer is $\frac{5}{3}$", I think what you mean is that you have constructed a model of the problem in Excel that suggests the answer is $\frac{5}{3}$. Your Excel model of the problem is wrong (as calculation by hand of the first few terms of of the sum shows). So the issue is in your Excel model. Debugging your Excel model is off topic for MSE.

Rob Arthan
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