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Use the ratio test to determine the radius of convergence of each series. $$ \sum_{n=1}^{\infty} \frac{2^{3 n}(n !)^{3}}{(3 n) !} x^{n} $$

In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. \(x+x^{2}-x^{3}-x^{4}+x^{5}+x^{6}-x^{7}-x^{8}+\cdots\) (Hint: Group powers \(x^{4 k}, x^{4 k-1}\), etc. \()\)

In the following exercises, find the Maclaurin series of each function. $$ f(x)=\cos ^{2} x \text { using the identity } \cos ^{2} x=\frac{1}{2}+\frac{1}{2} \cos (2 x) $$

In the following exercises, express the sum of each power series in terms of geometric series, and then express the sum as a rational function. \(x-x^{2}-x^{3}+x^{4}-x^{5}-x^{6}+x^{7}-\cdots\) (Hint: Group powers \(x^{3 k}, x^{3 k-1}\), and \(\left.x^{3 k-2} .\right)\)

In the following exercises, find the Maclaurin series of each function. $$ f(x)=\sin ^{2} x \text { using the identity } \sin ^{2} x=\frac{1}{2}-\frac{1}{2} \cos (2 x) $$

Use the ratio test to determine the radius of convergence of each series. $$ \sum_{n=1}^{\infty} \frac{n !}{n^{n}} x^{n} $$

In the following exercises, compute the Taylor series of each function around \(x=1\). $$ f(x)=2-x $$

In the following exercises, find the Maclaurin series of \(F(x)=\int_{0}^{x} f(t) d t\) by integrating the Maclaurin series of \(f\) term by term. If \(f\) is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero. $$ F(x)=\int_{0}^{x} e^{-t^{2}} d t ; f(t)=e^{-t^{2}}=\sum_{n=0}^{\infty}(-1)^{n} \frac{2^{2 n}}{n !} $$

Use the ratio test to determine the radius of convergence of each series. $$ \sum_{n=1}^{\infty} \frac{(2 n) !}{n^{2 n}} x^{n} $$

In the following exercises, compute the Taylor series of each function around \(x=1\). $$ f(x)=x^{3} $$