mathematics

Śleszyński–Pringsheim theorem

Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem rather than just read about it. In short: In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.

Key takeaways

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

Reference excerpt

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century. It states that if n {\displaystyle n} is a positive integer and ( a n ) {\displaystyle (a_{n})} , ( b n ) {\displaystyle (b_{n})} are sequences of real numbers such that | b n | ≥ | a n | + 1 {\displaystyle |b_{n}|\geq |a_{n}|+1} for all n {\displaystyle n} , then

a 1 b 1 + a 2 b 2 + a 3 b 3 + ⋱ {\displaystyle {\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}

converges absolutely to a number x {\displaystyle x} satisfying | x | ≤ 1 {\displaystyle |x|\leq 1} , meaning that the series

x = ∑ n { A n B n − A n − 1 B n − 1 } , {\displaystyle x=\sum _{n}\left\{{\frac {A_{n}}{B_{n}}}-{\frac {A_{n-1}}{B_{n-1}}}\right\},}

where A n / B n {\displaystyle A_{n}/B_{n}} are the convergents of the continued fraction, converges absolutely.

Proof Recall that the n {\displaystyle n} th convergents of x {\displaystyle x} , which will be denoted by A n B n {\displaystyle {\tfrac {A_{n}}{B_{n}}}} in this article, can be computed from the following recurrence relation:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Śleszyński–Pringsheim theorem

Start with the simplest possible case. Write down what Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem

In research: Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem 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: Śleszyński–Pringsheim theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continued fractions, Theorems in real analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Śleszyński–Pringsheim theorem in simple terms?

Why does Śleszyński–Pringsheim theorem 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 Śleszyński–Pringsheim theorem?

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 Śleszyński–Pringsheim theorem.