Exam Questions - Taylor’s series
Exam Questions - Taylor’s series
Common Question Formats
- Determining the Taylor or Maclaurin series expansion of a given function up to a certain degree.
- Calculating the nth term of a Taylor or Maclaurin series and identifying the function being represented.
- Approximating a function’s value at a point close to a which is typically 0 (Maclaurin).
- Using the Remainder Theorem to provide an estimate of the difference between the true value of the function and its approximation using Taylor or Maclaurin series.
- Determining the interval of convergence for a series expressing a function.
Strategies for Addressing Questions
- When asked to expand a function into its Taylor or Maclaurin series, remember the basic formula and replace ‘a’ with ‘0’ for a Maclaurin series.
- If you need to find the nth term of a series, think of deriving or integrating a known series.
- Recollect the definitions and properties of functions to recognize which function a series represents.
- To approximate a function’s value, add up the relevant number of terms of the series. Be careful to use a sufficient number of terms to achieve a required accuracy.
- For Remainder Theorem questions, remember that the remainder term is the first omitted term multiplied by the maximum value of the (n+1)th derivative of the function on the relevant interval.
- To find the interval of convergence, remember that the series converges where the absolute value of the ratio of the (n+1)th term to the nth term is less than 1.
Potential Pitfalls
- Be careful with factorials when dealing with higher derivatives.
- Make sure to write the general term if you are asked for the complete series. Not all functions can be expressed as finite polynomials.
- Remainder estimates are based on the worst case scenario, not averages.
- For the interval of convergence, note that endpoints need to be checked individually.
- Don’t forget to consider the domain of the original function when determining the interval of convergence.
Remember, practice makes perfect. The more you work with Taylor’s series, the more acquainted you will become with recognising patterns and applying strategies!