Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper

How can you calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using nothing more than a pen and pencil? Simply typing this in any symbolic calculator will give us $1/81$. I could also possibly find this formula if I was actually looking at given numbers but I have never tried working backwards. By backwards, I mean to be given the summation formula and determine the convergent limit for it (if it exists).

So supposing that we did not know the limit for this summation formula was $1/81$ and that we do not have any software for assistance, how do we calculate this summation formula without having to take this summation to infinity?

• I think you mean 1/81. – Numbersandsoon Oct 19 '13 at 1:36
• Whoops. Right you are. Fixed. – user67527 Oct 19 '13 at 1:37
• You have an Arithmetico Geometric Series. – Happy Green Kid Naps Oct 19 '13 at 5:12

Hint:

$$f(x) = \frac{1}{1 - x} = \sum\limits_{k = 0}^{\infty} x^k$$

and so differentiating term-by-term,

$$\frac{1}{(1 - x)^2} = \sum\limits_{k = 0}^{\infty} k x^{k - 1} = \sum\limits_{n = 1}^{\infty} (n - 1) x^n$$

• Although a great hint, how can we use this to see that $f(x)$ converges to a certain limit, where in this case is $1/81$? – user67527 Oct 19 '13 at 1:39
• @user67527 I don't understand your question. $f(x)$ isn't converging to a certain limit - it's just a function. Try plugging in an appropriate $x$ value. – user61527 Oct 19 '13 at 1:40
• T. Bongers is using the geometric series summation formula (in case you missed it). Very nice construction, plug in the appropriate $x$ value in the infinite series and see what you get! – Numbersandsoon Oct 19 '13 at 1:43
• IF we did not know that $\sum\limits_{n=1}^\infty (n-1)/10^n=1/81$, or in other words we were looking at $\sum\limits_{n=1}^\infty (n-1)/10^n = ?$, how could we find this $1/81$? Does that make sense? – user67527 Oct 19 '13 at 1:44
• @T.Bongers: Almost at the end, we want to multiply by $x^2$, the switch from $k$ to $n$ would give $(n+1)$, not $(n-1)$. – André Nicolas Oct 19 '13 at 1:54

T. Bongers has provided a general method, but I hope the following can help you understand it better. Write $$\sum_{n=1}^{\infty}\frac{(n-1)}{10^n}=\frac{(1-1)}{10}+\frac{(2-1)}{10^2}+\frac{(3-1)}{10^3}+\cdots\tag{1}$$ Therefore, $$10\cdot\sum_{n=1}^{\infty}\frac{(n-1)}{10^n}=(1-1)+\frac{(2-1)}{10}+\frac{(3-1)}{10^2}+\cdots\tag{2}$$ Now subtract $(1)$ from $(2)$. As a result, we get \begin{align*} 9\cdot\sum_{n=1}^{\infty}\frac{(n-1)}{10^n}&=\frac{(2-1)-(1-1)}{10}+\frac{(3-1)-(2-1)}{10^2}+\frac{(4-1)-(3-1)}{10^3}+\cdots\\ &=\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+\cdots\\ &=\frac{1}{1-1/10}-1\\&=\frac{1}{9}\end{align*} Hence, $$\sum_{n=1}^{\infty}\frac{(n-1)}{10^n}=\frac{1}{9}\cdot\frac{1}{9}=\frac{1}{81}$$

We know that \begin{align} A&=\hphantom{1+}\frac1{10}+\frac1{10^2}+\frac1{10^3}+\frac1{10^4}+\dots\\ 10A&=1+\frac1{10}+\frac1{10^2}+\frac1{10^3}+\frac1{10^4}+\dots\\[4pt] 9A&=1 \end{align} So $\frac1{10}+\frac1{10^2}+\frac1{10^3}+\frac1{10^4}+\dots=A=\frac19$.

Likewise, \begin{align} \color{#C00000}{\frac1{10}}A&=\frac1{10^2}+\frac1{10^3}+\frac1{10^4}+\frac1{10^5}+\dots\\ \color{#C00000}{\frac1{10^2}}A&=\hphantom{\frac1{10^2}+}\frac1{10^3}+\frac1{10^4}+\frac1{10^5}+\dots\\ \color{#C00000}{\frac1{10^3}}A&=\hphantom{\frac1{10^2}+\frac1{10^3}+}\frac1{10^4}+\frac1{10^5}+\dots\\ \color{#C00000}{\frac1{10^4}}A&=\hphantom{\frac1{10^2}+\frac1{10^3}+\frac1{10^4}+}\frac1{10^5}+\dots\\ &\vdots\quad\text{summing the equations above}\\ \color{#C00000}{A}\hphantom{A}A&=\frac1{10^2}+\frac2{10^3}+\frac3{10^4}+\frac4{10^5}+\dots \end{align} So $\frac1{10^2}+\frac2{10^3}+\frac3{10^4}+\frac4{10^5}+\dots=A^2=\frac1{81}$.

$$\sum\limits_{n=1}^\infty \frac{n-1}{10^n} = \sum\limits_{n=1}^\infty \sum\limits_{i=1}^{n-1} \frac{1}{10^n}= \sum\limits_{i=1}^\infty \sum\limits_{n=i+1}^\infty \frac{1}{10^{n}} = \sum\limits_{i=1}^\infty\frac{10^{-(i+1)}}{1-\frac{1}{10}}=\frac{1}{9}\left(\frac{1}{10}+\frac{1}{100}+\cdots\right)=\frac{1}{81}$$

As a picture: 