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Magazine Problem 33
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{x} ; a=1 \text { (Hint: } \left.\frac{1}{x}=\frac{1}{1-(1-x)}\right) $$
Problem 34
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1-x^{2}} ; a=0 $$
Problem 35
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{x}{1-x^{2}} ; a=0 $$
Problem 36
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1+x^{2}} ; a=0 $$
Problem 37
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{x^{2}}{1+x^{2}} ; a=0 $$
Problem 38
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{2-x} ; a=1 $$
Problem 39
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1-2 x} ; a=0 $$
Problem 40
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{1}{1-4 x^{2}} ; a=0 $$
Problem 41
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{x^{2}}{1-4 x^{2}} ; a=0 $$
Problem 42
Given that \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}\) with convergence in \((-1,1),\) find the power series for each function with the given center \(a\), and identify its interval of convergence. $$ f(x)=\frac{x^{2}}{5-4 x+x^{2}} ; a=2 $$
