mathematics

Ramanujan–Sato series

Ramanujan–Sato 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 Ramanujan–Sato series rather than just read about it. In short: In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, 1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{k=0}^{\infty }{\frac {(4k)!}{k!^{4}}}{\frac {26390k+1103}{396^{4k}}}} to the form 1 π = ∑ k = 0 ∞ s ( k ) A k + B C k {\displaystyle {\frac {1}{\pi }}=\sum _{k=0}^{\infty }s(k){\frac {Ak+B}{C^{k}}}} b…

Key takeaways

Ramanujan–Sato 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 Ramanujan–Sato series to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ramanujan–Sato series from memory before moving on to harder problems.

Reference excerpt

In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as,

1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{k=0}^{\infty }{\frac {(4k)!}{k!^{4}}}{\frac {26390k+1103}{396^{4k}}}}

to the form

1 π = ∑ k = 0 ∞ s ( k ) A k + B C k {\displaystyle {\frac {1}{\pi }}=\sum _{k=0}^{\infty }s(k){\frac {Ak+B}{C^{k}}}}

by using other well-defined sequences of integers s ( k ) {\displaystyle s(k)} obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} , and A , B , C {\displaystyle A,B,C} employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only in 2012 that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} , while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators. Levels 1–4A were given by Ramanujan (1914), level 5 by H. H. Chan and S. Cooper (2012), 6A by Chan, Tanigawa, Yang, and Zudilin, 6B by Sato (2002), 6C by H. Chan, S. Chan, and Z. Liu (2004), 6D by H. Chan and H. Verrill (2009), level 7 by S. Cooper (2012), part of level 8 by Almkvist and Guillera (2012), part of level 10 by Y. Yang, and the rest by H. H. Chan and S. Cooper. The notation jn(τ) is derived from Zagier and Tn refers to the relevant McKay–Thompson series.

Level 1 Examples for levels 1–4 were given by Ramanujan in his 1917 paper. Given q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} as in the rest of this article. Let,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ramanujan–Sato series

Start with the simplest possible case. Write down what Ramanujan–Sato 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 Ramanujan–Sato 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 Ramanujan–Sato 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 Ramanujan–Sato series

In research: Ramanujan–Sato 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 Ramanujan–Sato 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: Ramanujan–Sato series is common in secondary-school and first-year university syllabi. It links to neighbouring topics Pi algorithms, Series (mathematics), Srinivasa Ramanujan, so understanding it makes those chapters shorter.
In everyday life: Look for Ramanujan–Sato 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 Ramanujan–Sato series in 20 minutes

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

Frequently asked questions

What is Ramanujan–Sato series in simple terms?

In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, 1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{k=0}^{\infty }{\frac {(4k)!}{k!^{4}}}{\frac {26390k+1103}{396^{4k}}}} to the for…

Why does Ramanujan–Sato 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 Ramanujan–Sato 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 Ramanujan–Sato series.