How to sum $nC^{n-1}$ from 0 to infinity?

Question

Where C is a const. e.g. sum $n(.5)^{n-1}$, OR sum $n(2)^{n-1}$ from 0 to infinity. Thank you.

See this similar question for ideas. – David Mitra May 06 '13 at 12:02 — David Mitra, May 06 '13 at 12:02

Ragib Zaman · Accepted Answer · 2013-05-06T12:03:51.290

2

Differentiate both sides of $\displaystyle\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n.$

In general, if we have a function $f(x)$ which is given by a power series $f(x) =\displaystyle \sum_{n=0}^{\infty} a_n x^n$ with radius of convergence $R$ then $f$ is differentiable, and $f'(x) =\displaystyle \sum_{n=0}^{\infty} (n+1)a_{n+1} x^{n}.$ This is because for every $R>\epsilon>0,$ the Weierstrass M-test gives that $\displaystyle f_m(x) = \sum_{n=0}^m a_n x^n$ converges uniformly to $f(x)$ on $[-R+\epsilon, R-\epsilon].$ If $g_n \to g$ uniformly on $[a,b]$ and $g'_n \to h$ on $[a,b]$ then $g$ is differentiable on $(a,b)$ and $g'=h$ (this is a basic theorem in real analysis) so we have justified why we can differentiate power series term by term.

edited May 06 '13 at 12:03

answered May 06 '13 at 11:48

Ragib Zaman

35,127

Why the sum on the right from 0 to infinity could be diff.? – RHS May 06 '13 at 11:52
@RHS See my edited answer. – Ragib Zaman May 06 '13 at 12:05

How to sum $nC^{n-1}$ from 0 to infinity?

1 Answers1