A Series Summation Proof (Without Mathematical Induction)

一道级数求和证明题(非数学归纳法) 苏剑林 · 2009-08-02 · Mathematics series summation mathematical induction

Today on Mathematical Research and Development Forum I came across a problem:

$$\sum_{j=0}^{j=n} (jx^j)={nx^{n+2}-(n+1)x^{n+1}+x}/{(x-1)^2}$$

This problem is really just asking for the summation formula of $x+2x^2+3x^3+...+nx^n$.

Using mathematical induction would have been quite straightforward (induction is easy for proving something, but no good for deriving it in the first place), but the problem stipulates that induction cannot be used. So we're left with the following approach. more

Let's rewrite it as:

$$\begin{aligned}x(1+2x+3x^2+...+nx^{n-1}) \\ \Rightarrow x[1+x+x^2+...+x^{n-1}+x(1+x+x^2+...+x^{n-2})+...+x^{n-1}] \\ \Rightarrow x[{x^n-1+x(x^{n-1}-1)+x^2(x^{n-2}-1)+...+x^{n-1}(x-1)}/{x-1}] \\ \Rightarrow x[{nx^n-(1+x+x^2+...+x^{n-1})}/{x-1}] \\ \Rightarrow x[{nx^n-{x^n-1}/{x-1}}/{x-1}]\end{aligned}$$

$\Rightarrow x[{nx^{n+1}-(n+1)x^n+1}/{(x-1)^2}]

The rest I won't bother writing out. No need to say more, right?