A sequence is an ordered list of numbers; a series is the sum of a sequence. The jump from finite to infinite sums is not just a technicality — it is the mathematical scaffolding for present-value formulas (geometric series), yield-curve fitting (power series), and the approximation of normal CDF integrals and option prices (Taylor polynomials). The payoff of this topic is power series and Taylor series, which represent functions as infinite polynomials: , . These expansions give bond-yield approximations (duration and convexity are Taylor coefficients), small-vol expansions of option prices, and the derivation of continuously compounded interest from the discrete limit.
By the end of this topic you should be able to:
- determine whether a sequence converges and apply the Squeeze Theorem and Monotone Convergence Theorem;
- test infinite series for convergence using all standard tests (Integral, Comparison, Ratio, Root, Alternating);
- find the radius and interval of convergence of a power series;
- represent functions as power series by differentiation, integration, and substitution;
- write the Taylor and Maclaurin series for standard functions and estimate the error with Taylor's Inequality.
In this lesson
- Contents
- The big picture
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