EC-18.5 Sequences, Series, and Approximating a Function

What this is and why it exists

This topic is the reason linearised models are allowed rather than merely convenient.

Every small-signal model in the devices course is a Taylor series cut off after one term. Every linearised model in the control course is the same. An engineer who does not know that cannot say when the linearisation stops being safe. Nor can they say when a design has left the region where its own model applies.

It is also the groundwork for the Fourier series taught in the transforms course. That series is an infinite sum, and whether it converges, and to what, is the entire question there.

The vocabulary

• Sequence — an ordered list of numbers, indexed by the counting numbers.
• Convergent sequence — one whose terms settle towards a single value and stay near it.
• Series — the sum of the terms of a sequence.
• Partial sum — the total of the first several terms.
• Geometric series — one where each term is a fixed multiple of the one before.
• Comparison, ratio and root tests — three ways to decide whether a series converges.
• Absolute convergence — convergence that survives replacing every term by its size.
• Conditional convergence — convergence that does not survive that replacement.
• Power series — a series in whole-number powers of the input.
• Radius of convergence — the distance within which a power series is valid.
• Taylor series — a power series built from the derivatives of a function at a position.
• Remainder term — the bound on the error made by stopping the series early.

The mental model

A sequence converges when its terms settle. That is not an abstraction invented for the subject. Sampled measurements form a sequence. So do the successive outputs of Newton's method, and so do the successive partial sums of any series.

A series is defined as the limit of its partial sums, which makes it a sequence question wearing different clothes. Add the first term, then the first two, then the first three. If those totals settle, the series converges to what they settle on.

The geometric series is the one case worth memorising. Each term is a fixed multiple of the one before. When the size of that multiplier is less than one the total settles; when it is one or more the total grows without bound. This reappears wherever a repeated reflection, a repeated delay or a repeated round trip is summed.

The convergence tests are three routes to one question, and choosing between them is a matter of the shape of the terms.

• Comparison when the terms resemble something already known.
• Ratio when consecutive terms have a clean quotient, which is common with factorials and powers.
• Root when the terms carry a power of the index.

Absolute against conditional convergence is worth one paragraph. A series converges absolutely when it still converges after every term is replaced by its size. Such a series can be rearranged freely. A conditionally convergent series can be rearranged into a different total, which is a fact worth knowing about even if you never rely on it.

Power series come next. Inside a radius, a power series behaves like an ordinary polynomial: it can be added, differentiated and integrated term by term. Outside that radius it means nothing at all. Using an expansion beyond where it converges produces confident nonsense rather than an obvious failure, which is what makes the radius worth checking.

The Taylor series is the payoff. Near a position, a sufficiently smooth function is approximated by a polynomial built from its derivatives there. The constant term matches the value. The linear term matches the slope. The quadratic term matches the curvature, and so on.

What turns that into an engineering tool is the remainder term. It bounds the error you accepted by stopping early. Without a bound you have an approximation; with a bound you have an engineering result.

Cutting the series off after the linear term is the single most common act in this area. It produces the small-angle approximation, where the sine of a small angle is taken as the angle itself. It produces every small-signal device model, where a curve is replaced by its tangent at an operating point. It produces every linearised control model.

The practical question is always the same: how far from the position does the approximation survive? The next term in the series answers it. If the quadratic term is small compared with the linear one over the range you care about, the linear model is safe there. If it is not, the model has left its region and the design built on it is unsupported.

What you should now be able to explain or do

• Decide whether a sequence converges, and connect that to the partial sums of a series.
• Recognise a geometric series and say when it settles.
• Choose a convergence test from the shape of the terms and apply it.
• Explain the difference between absolute and conditional convergence and why it matters.
• Find a radius of convergence and say what happens outside it.
• Build a Taylor approximation, cut it off deliberately, and bound the error you accepted.

Check yourself