EC-18.5 Sequences, Series, and Approximating a Function
You can decide whether an infinite sum converges, replace an awkward function by a polynomial, and state honestly how much error that replacement cost you.
Before:nothing requiredUnlocks:EC-3. Differential Equations
This topic is the reason small-angle approximations and first-order models are allowed rather than merely convenient. Every linearised model in the later subjects is a Taylor series cut off after one term, and an engineer who does not know that cannot say when the linearisation stops being safe. It is also the groundwork for the Fourier series taught in the transforms course, which is an infinite sum whose convergence is the entire question.
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Sequences, and what it means for one to converge
A sequence converges when its terms settle towards a single value and stay near it. Sampled measurements and the successive outputs of an iterative method are both sequences, so this is not an abstraction for its own sake.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · CourseSeries, partial sums, and the geometric series
An infinite sum is defined as the limit of its partial sums, and the geometric series is the one case with a formula worth memorising. It reappears in signal processing wherever a repeated reflection or a repeated delay is summed.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseTests for convergence, and absolute against conditional
Comparison, ratio and root tests each answer the same question by a different route, and choosing the right one is a matter of the shape of the terms. A series that converges only conditionally can be rearranged into a different answer, which is worth knowing about.
MIT 18.01SC: Single Variable Calculus · CoursePower series, and the interval where one is valid
A power series behaves like an ordinary polynomial inside a radius and is meaningless outside it. Using an expansion beyond where it converges is a mistake that produces confident nonsense rather than an obvious failure.
Taylor and Maclaurin series, and the remainder term
Any sufficiently smooth function can be approximated near a point by a polynomial built from its derivatives there. The remainder term is what makes it an engineering tool instead of a curiosity, because it bounds the error you accepted.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseSmall-angle and first-order approximations, and where engineering leans on them
Cutting a Taylor series off after the linear term is what produces every small-signal model and every linearised control model in the later subjects. Knowing the next term tells you how far from the operating point the approximation survives.
MIT 18.01SC: Single Variable Calculus · Course
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