If f(x)=∑n=0∞cnxnf(x) = \sum_{n=0}^{\infty} c_n x^nf(x)=∑n=0∞cnxn and f′(x)=∑n=1∞ncnxn−1f'(x) = \sum_{n=1}^{\infty} n c_n x^{n-1}f′(x)=∑n=1∞ncnxn−1, find the power series for f′(x)f'(x)f′(x) where f(x)=11−xf(x) = \frac{1}{1-x}f(x)=1−x1.
∑n=0∞nxn−1\sum_{n=0}^{\infty} n x^{n-1}∑n=0∞nxn−1
∑n=1∞nxn−1\sum_{n=1}^{\infty} n x^{n-1}∑n=1∞nxn−1
∑n=0∞(n+1)xn\sum_{n=0}^{\infty} (n+1) x^n∑n=0∞(n+1)xn
∑n=1∞n2xn\sum_{n=1}^{\infty} n^2 x^n∑n=1∞n2xn