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Magazine Problem 22
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow\left(1-x^{2}\right)^{-1 / 2} $$
Problem 22
In each of Problems 17 through 24, find all the values of \(x\) for which the given power series converges. $$ \sum_{n=0}^{\infty} \frac{\left(2 n^{2}+2 n+1\right) x^{n}}{2^{n}(n+1)^{3}} $$
Problem 23
In each of Problems 17 through 24, find all the values of \(x\) for which the given power series converges. $$ \sum_{n=1}^{\infty} \frac{\log (n+1) 2^{n}(x+1)^{n}}{n+1} $$
Problem 23
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow\left(1+x^{2}\right)^{-1 / 2} $$
Problem 24
In each of Problems 17 through 24, find all the values of \(x\) for which the given power series converges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(\log n) 2^{n} x^{n}}{3^{n} n^{2}} $$
Problem 24
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow(1-x)^{-2} $$
Problem 25
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow(1-x)^{-3} $$
Problem 26
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow \arcsin x $$
Problem 27
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow\left(1-x^{2}\right)^{-3} $$
Problem 28
In each of Problems 22 through 29 find the Taylor expansion about \(a=0\) and prove its validity. $$ f: x \rightarrow \arctan \left(x^{2}\right) $$
