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 \(\arctan x\).