mathematics

Function series

Function series is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Function series rather than just read about it. In short: In calculus, a function series is a series where each of its terms is a function, not just a real or complex number. Examples Examples of function series include ordinary power series, Laurent series, Fourier series, Liouville-Neumann series, formal power series, and Puiseux series.

Key takeaways

Function series belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Function series to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Function series from memory before moving on to harder problems.

Reference excerpt

In calculus, a function series is a series where each of its terms is a function, not just a real or complex number.

Examples Examples of function series include ordinary power series, Laurent series, Fourier series, Liouville-Neumann series, formal power series, and Puiseux series.

Convergence There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere. Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit. The Weierstrass M-test is a useful result in studying convergence of function series.

Worked examples

Example 1 — a first encounter with Function series

Start with the simplest possible case. Write down what Function series claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Function series before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Function series ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Function series

In research: Function series appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Function series in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Function series is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical analysis, Series (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Function series outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Function series in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Function series means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Function series out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Function series in simple terms?

In calculus, a function series is a series where each of its terms is a function, not just a real or complex number. Examples Examples of function series include ordinary power series, Laurent series, Fourier series, Liouville-Neumann series, formal power series, and Puiseux series.

Why does Function series matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Function series?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Function series.