MATHEMATICS S6 UNIT 3: Taylor and Maclaurin’s Expansions

By Mukongori Categories: S6 MATHEMATICS

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About Course

COLIN MACLAURIN (1698–1746)
The development of power series to represent functions is credited to the combined work of many seventeenth and eighteenth century mathematicians. Gregory, Newton, John and James Bernoulli, Leibniz, Euler, Lagrange, Wallis, and Fourier all contributed to this work. However, the two names that are most commonly associated with power series are Brook Taylor (1685–1731) and Colin Maclaurin.

Find a Taylor or Maclaurin series for a function.
Find a binomial series.
Use a basic list of Taylor series to find other Taylor series.

Find the sum of a given series.
Find the Taylor series of a given function.
Find the Maclaurin series of a given function.
Use Maclaurin series to;
calculate limits,
approximate the values of some constants,
approximate an irrational number,
approximate logarithmic number,
approximate trigonometric number of an angle,
approximate the roots of a given equation.

Course Content

Finite series

Infinite series

Test for convergence of series

Power series

Taylor and Maclaurin series

Taylor series by using Maclaurin series

Calculation of limits

Estimation of the number e

Estimation of the number π

Estimation of trigonometric number of an angle

Estimation of an irrational number

Estimation of a natural logarithm number

Estimation of roots of equations

Final Exam

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A course by

Functions: A thorough understanding of what a function is, its domain, range, and various types of functions (polynomial, exponential, logarithmic, trigonometric).
Algebra: Proficiency in algebraic manipulation, including simplifying expressions, solving equations, and working with exponents and factorials. Factorials (n!) are integral to the Taylor and Maclaurin formulas.
Graphing: Ability to sketch and interpret graphs of various functions. This helps in visualizing how Taylor polynomials approximate the original function.
Limits: Understanding the concept of a limit is fundamental, as series involve infinite sums and convergence.
Derivatives: This is the most crucial prerequisite. Taylor and Maclaurin series are defined using the derivatives of a function at a specific point.
Continuity and Differentiability: Understanding that a function must be sufficiently differentiable at the point of expansion for its Taylor series to exist. For a function to have a Taylor series, it must be infinitely differentiable (smooth) at the point of expansion.
Antiderivatives and Indefinite Integrals: While not directly used in the definition of Taylor/Maclaurin series, the concept of integration is essential for understanding how the remainder term in Taylor's Theorem is derived (often through repeated integration by parts).
Definite Integrals: Understanding how definite integrals relate to areas under curves and accumulation.
Sequences: Understanding what a sequence is and how to determine if it converges or diverges.
Series: This is critical ( The definition of an infinite series and partial sums, Power Series: Taylor and Maclaurin series are a special type of power series. Understanding the general form of a power series, Interval and Radius of Convergence: Being able to find the interval and radius for which a power series converges.)

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