Manipulating Taylor Series

Now that we’ve looked at operations involving Taylor series, complete the questions below for our next lesson.

1. Find the Taylor polynomial of degree 3, centred at $0$, for $e^x\sin 2x$.
2. Find the Taylor series, centred at $0$, for $\sin x+\cos x$.
3. Us the series
   $$\frac{1}{1-x}=1+x+x^2+\cdots+x^n+\cdots$$ to find the Taylor series for
   1. $\displaystyle{\frac{1}{1-2x}}$
   2. $\displaystyle{\frac{1}{1+x}}$
   3. $\displaystyle{\frac{1}{1+x^2}}$
4. Determine the interval of convergence for each series in questions 1 to 3.
5. Use the Taylor series for $\displaystyle{\frac{1}{1+x^2}}$ to find the Taylor series centred at $0$ for $\arctan x$. Determine the interval of convergence for the Taylor series at $0$ for $\arctan x$.