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Could you please explain how come: $$\sum_{i=1}^\infty(0.5)^{i+1}(i+1)=2$$

How come the answer is not inf? And why 2?

Any help will be appreciated.

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See this similar question for ideas. –  David Mitra Apr 19 '13 at 11:55

2 Answers 2

up vote 2 down vote accepted

We know for $|r|<1$, $$\sum_{1\le i<\infty}r^{i+1}=r^2\cdot\frac1{1-r}=\frac{r^2}{1-r}$$

$$\text{Now, }\frac{d\left(\sum_{1\le i<\infty}r^{i+1}\right)}{dr}=\sum_{1\le i<\infty}\frac{d r^{i+1}}{dr}=\sum_{1\le i<\infty}(i+1)r^i$$

$$\text{and }\frac{d\left(\frac{r^2}{1-r}\right)}{dr}=\frac{2r-r^2}{(1-r)^2}$$

$$\implies \sum_{1\le i<\infty}(i+1)r^i=\frac{2r-r^2}{(1-r)^2}$$

$$\implies \sum_{1\le i<\infty}(i+1)r^{i+1}=\frac{r(2r-r^2)}{(1-r)^2}$$

Alternatively, using Arithmetico-geometric series, we can show, $$\sum_{1\le i<\infty}(i+c)r^{i+1}=\frac{(c+1)r^2-c\cdot r^3}{(1-r)^2}$$ where $c$ is any finite number

Let $S=\sum_{1\le i<\infty}(i+c)r^{i+1}$

Multiply either sides by the common ratio $r,$ and subtract from the original

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I'm not sure what you did here and why. Can you please explain? –  Amanda Apr 19 '13 at 11:31
    
@Mankochka, do you know Infinite Geometric Series ( en.wikipedia.org/wiki/Geometric_series#Proof_of_convergence) with common ratio=$r$? If yes, the apply derivative wrt $r$ –  lab bhattacharjee Apr 19 '13 at 11:37
    
I understood the proof you linked to, but as it doesn't mention derivatives I don't understand how to apply it to my example, and what are the steps you performed. –  Amanda Apr 19 '13 at 13:02
    
@Mankochka, please find the edited answer –  lab bhattacharjee Apr 19 '13 at 16:02
    
Finally got it, thanks :) –  Amanda Apr 19 '13 at 16:34

You wonder why the sum

$$\sum_{k=2}^{\infty} k \left ( \frac{1}{2}\right)^k$$

does not blow up, presumably because of the factor of $k$ in the sum which grows linearly to $\infty$. The answer, very roughly, is because the factor of $(1/2)^k$ decreases much more quickly than $k$ increases.

Let's look at a few terms to see how this works. The first few terms in the sum look like

$$2 \left( \frac{1}{2}\right)^2 + 3 \left( \frac{1}{2}\right)^3 + 4 \left( \frac{1}{2}\right)^4 + \ldots = \frac{2}{4} + \frac{3}{8} + \frac{4}{16}+ \ldots$$

The terms here are already getting smaller at $k=2$. Now let's double the index:

$$\ldots+\frac{8}{2^8}+\ldots+\frac{16}{2^{16}}+\ldots+\frac{32}{2^{32}}+\ldots$$

You should see that, while the numerator doubles, i.e., exponent increases by $1$, the denominator is squared, i.e., exponent is doubled.

I hope you see that the terms in the series rapidly vanish as $k \rightarrow \infty$. By comparison with far more slowly varying sequences whose corresponding series converge (e.g., $1/k^2$), this series converges.

It converges to $3/2$ for reasons summarized by @lab above.

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I wish I could accept both answers, thanks :) –  Amanda Apr 19 '13 at 16:35

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