Power series

Is $\displaystyle \sum_{n=1}^\infty {(0.1)}^n x^n$ a convergent power series? If so, find the radius of convergence.

Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n} x^n$ a convergent power series? If so, find the radius of convergence.

Using the geometric series, expand $\frac{1}{1-x}$ around $x_0=2$. For what $x$ does the series converge?

Find the Taylor series for $x^7 e^x$ around $x_0 = 0$.

Imagine $f$ and $g$ are analytic functions such that $f^{(k)}(0) = g^{(k)}(0)$ for all large enough $k$. What can you say about $f(x)-g(x)$?

Reindex the following series to have the powers be $x^k$.

1. $\displaystyle \sum_{k=3}^\infty k(k-1)x^{k+3}$
2. $\displaystyle \sum_{k=1}^\infty (k+1)x^{k-1}$
3. $\displaystyle \sum_{k=5}^\infty 2kx^{k-2}$