Examples and practice
Maclaurin series and approximation
Back to lessonPractice section 1 · Questions 1–6
Expand a logarithm after changing its argument · Transform the interval along with the series argument · +4 more
- Expand a logarithm after changing its argument
- Transform the interval along with the series argument
- Rewrite before choosing ascending powers
- Practise finite and nonterminating binomial expansions
- Intersect the intervals after using partial fractions
- Choose a valid substitution and assess the resulting estimate
Practice section 2 · Questions 7–12
Build a fourth-degree series from a differential equation · Differentiate a tangent expansion to obtain a secant-square series · +4 more
- Build a fourth-degree series from a differential equation
- Differentiate a tangent expansion to obtain a secant-square series
- Verify an implicit logarithm expansion by a second method
- Practise a differential relation and a logarithm of secant
- Expand shifted angles before making small-angle approximations
- Approximate a shifted tangent using an exact identity
Practice section 3 · Questions 13–18
Track the order of an approximation through division · Normalize a square root before using the binomial series · +4 more
Practice section 4 · Questions 19–24
Factor out a negative power before expanding · Match a product-series coefficient · +4 more
Practice section 5 · Questions 25–30
Transform an exponential series without differentiating again · Match trigonometric and exponential series · +4 more
Practice section 6 · Questions 31–36
Use a derivative identity for an even exponential composition · Verify an exponential–sine series by two methods · +4 more
- Use a derivative identity for an even exponential composition
- Verify an exponential–sine series by two methods
- Differentiate the logarithmic series to obtain a logistic expansion
- Match two series without matching their cubic terms
- Recover an inverse-cosine tangent from its Maclaurin coefficients
- Compose a square-root series with an inverse tangent
Practice section 7 · Questions 37–39
Compare a cubic approximation with the actual function · Use the printed half-unit initial value in a derivative recurrence · +1 more