In previous videos I have shown you the series expansions of the following functions using Maclaurin’s series.

Useful identities:

  • {e^x} \equiv 1 + x + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + ... + \dfrac{{{x^r}}}{{r!}} + ...{\text{                               valid for all values of }}x
  • \sin x \equiv x - \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^5}}}{{5!}} - \dfrac{{{x^7}}}{{7!}} + ... + \dfrac{{{{\left( { - 1} \right)}^r}{x^{2r + 1}}}}{{\left( {2r + 1} \right)!}} + ...{\text{            valid for all values of }}x
  • \cos x \equiv 1 - \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^4}}}{{4!}} - \dfrac{{{x^6}}}{{6!}} + ... + \dfrac{{{{\left( { - 1} \right)}^r}{x^{2r}}}}{{\left( {2r} \right)!}} + ...{\text{               valid for all values of }}x
  • \ln \left( {1 + x} \right) \equiv x - \dfrac{{{x^2}}}{2} + \dfrac{{{x^3}}}{3} - \dfrac{{{x^4}}}{4} + ... + \dfrac{{{{\left( { - 1} \right)}^{r - 1}}{x^{2r}}}}{r} + ...{\text{      }} - 1 < x \le 1

In this video I show you how we can use these series to to derive further series.

Further series using Maclaurins Series : ExamSolutions Maths Revision - youtube Video