Consider f(x)=∑n=0∞xnf(x) = \sum_{n=0}^{\infty} x^nf(x)=∑n=0∞xn. What is the power series for its derivative f′(x)f'(x)f′(x) for ∣x∣<1|x| < 1∣x∣<1?
∑n=0∞nxn−1\sum_{n=0}^{\infty} n x^{n-1}∑n=0∞nxn−1
∑n=1∞xn\sum_{n=1}^{\infty} x^n∑n=1∞xn
∑n=1∞nxn−1\sum_{n=1}^{\infty} n x^{n-1}∑n=1∞nxn−1
∑n=0∞xn−1\sum_{n=0}^{\infty} x^{n-1}∑n=0∞xn−1