Power Series

Use the Maclaurin series for $\frac{1}{1-x}$to find a power series representation for $\frac{x^2}{{(1-x^3)}^2}$

Use the Maclaurin series for $\frac{1}{1-x}$ to find power series representations for $\frac{1}{1+x}$ , $\ln (1+x)$ , $\int \ln (1+x)dx$ , and ${\int }_0^2\ln (1+x)dx$.

Use the Maclaurin series for cos x to find series representations for $\cos \left(x^3\right),\int \cos \left(x^3\right)dx, {\int }_0^1\cos \left(x^3\right)dx$

Find the Maclaurin series for $e^{-x}$, and use it to approximate $\frac{1}{\sqrt{e}}$ to within 0.001 of its value. How many terms would you need to approximate $\frac{1}{\sqrt{e}}$ to within ${10}^{-6}$ of its value? 

ind the Maclaurin series for $e^{-x^2}$ , and use it to approximate ${\int }_0^1{e}^{-x^2}dx$ to within 0.001 of its value. How many terms would you need to approximate the integral to within ${10}^{-6}$ of its value? 

Find the Maclaurin series for $\sin \left(x^2\right)$, and use it to approximate ${\int }_0^3\sin \left(x^2\right)dx$ to within 0.001 of its value. How many terms would you need to approximate the integral to within ${10}^{-6}$ of its value?