91Ó°ÊÓ

For each natural number \(n\) and each number \(x,\) define $$ f_{n}(x)=\frac{1-|x|^{n}}{1+|x|^{n}} $$ Find the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) to which the sequence \(\left\\{f_{n}: \mathbb{R} \rightarrow \mathbb{R}\right\\}\) converges pointwise.

Determine the domains of convergence of each of the following power series: a. \(\sum_{k=1}^{\infty} \frac{x^{k}}{k 5^{k}}\) b. \(\sum_{k=1}^{\infty} k ! x^{k}\) c. \(\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k-1}}{(2 k+1) !}\)

Use the Cauchy Convergence Criterion for Series to provide another proof of the Alternating Series Test.