Prove:
The sequence $\displaystyle b_n=1+2x+3x^2+...+nx^{n-1}, |x|<1$ converges to $\displaystyle (1-x)^{-2}$
I don't even know what method to use to show:
1. that this converges, and
2. that it converges to that value.
Any help with this one?
Prove:
The sequence $\displaystyle b_n=1+2x+3x^2+...+nx^{n-1}, |x|<1$ converges to $\displaystyle (1-x)^{-2}$
I don't even know what method to use to show:
1. that this converges, and
2. that it converges to that value.
Any help with this one?
$\displaystyle S=1+2x+3x^2+...+nx^{n-1}+...\implies \frac{S-1}{x}=2+3x+4x^2+...+nx^{x-2}+...$
Now subtract $\displaystyle 1+x+x^2+...+x^{n-2}+...=\frac{1}{1-x}$ from both sides:
$\displaystyle \frac{S-1}{x}-\frac{1}{1-x}=1+2x+3x^2+...+(n-1)x^{n-2}+...$
But the right side of the equation now equals $\displaystyle S$, so solve for it in the equation below.
$\displaystyle \frac{S-1}{x}-\frac{1}{1-x}=S$
Obviously, this method is not as fast as differentiation.