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While during physics I encountered a sum I couldn't evaluate: $$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$ Is there a particular formula for this sum and does it converges?

Martin Sleziak
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PunkZebra
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    Are you aware that an infinite series is a limit of its partial sums (the sum of the first $n$ terms, as $n \rightarrow \infty$)? If you sum the first 5 terms you get .6875, which might not seem so clear in terms of the trend, but the sum of the first 10 terms is .66601... and the sum of the first 20 terms is .66666603..., so the obvious conjecture to make is that $S = 2/3$. My point is not that this numerical experimentation is going to help you derive the exact answer, but it does make it quite clear what to guess the answer should be, and that's an important skill. – KCd Jul 04 '14 at 18:45
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    How did this appear in physics? If you don't mind me asking lol – Ethan Splaver Jul 04 '14 at 18:51

5 Answers5

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Hint: This is a geometric series with ratio $-\dfrac{1}{2}$.

Please see the wikipedia article on geometric series.

In general we have that $1+r+r^2+\ldots=\dfrac{1}{1-r}$ given $|r|<1$.

Peter Woolfitt
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We sum the geometric series:

$$S=\sum_{n=0}^\infty\left(\frac{-1}{2}\right)^n=\frac{1}{1-\left(\frac{-1}{2}\right)}=\frac23$$

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Another easy way is to note that $2S=2-S$, whence $S=\frac{2}{3}$

Nicky Hekster
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Using techniques that are frequently used by physicists: \begin{align*} 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots &= \left( 1 - \frac{1}{2} \right) + \left( \frac{1}{4} - \frac{1}{8} \right) + \cdots \\ &= \frac{1}{2} + \frac{1}{8} + \cdots \\ &= \frac{1}{2} \left( 1 + \frac{1}{4} + \frac{1}{16} + \cdots \right) \\ &= \frac{1}{2} \frac{1}{1-\frac{1}{4}} \\ &= \frac{1}{2-\frac{1}{2}} \\ &= \frac{1}{3/2} \\ &= \frac{2}{3} \end{align*}

Note: There is no error in the parenthesization in the first line. We know the series converges (alternating series test), so every subsequence of the partial sums converges to the same thing, so this particular subsequence of the partial sums converges to the same thing.

Eric Towers
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$$S=1 - \frac {1}{2} + \frac {1}{4} - \frac {1}{8} ...$$ Let this be (1)

Multilply both sides by $\frac {1}{2}$ we get

$$\frac{S}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{8} ...$$ and this be (2)

Now Add (1) and (2)

$$\frac{3}{2} \cdot S=1$$

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

MonK
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