mathematics

Summation by parts

Summation by parts 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 Summation by parts rather than just read about it. In short: In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.

Key takeaways

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

Reference excerpt

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.

Statement Suppose { f k } {\displaystyle \{f_{k}\}} and { g k } {\displaystyle \{g_{k}\}} are two sequences. Then,

∑ k = m n f k ( g k + 1 − g k ) = ( f n + 1 g n + 1 − f m g m ) − ∑ k = m n g k + 1 ( f k + 1 − f k ) . {\displaystyle \sum _{k=m}^{n}f_{k}(g_{k+1}-g_{k})=\left(f_{n+1}g_{n+1}-f_{m}g_{m}\right)-\sum _{k=m}^{n}g_{k+1}(f_{k+1}-f_{k}).}

Using the forward difference operator Δ {\displaystyle \Delta } , it can be stated more succinctly as

∑ k = m n f k Δ g k = ( f n + 1 g n + 1 − f m g m ) − ∑ k = m n g k + 1 Δ f k , {\displaystyle \sum _{k=m}^{n}f_{k}\Delta g_{k}=\left(f_{n+1}g_{n+1}-f_{m}g_{m}\right)-\sum _{k=m}^{n}g_{k+1}\Delta f_{k},}

Or, equivalently, using backward difference operator ∇ {\displaystyle \nabla } :

∑ k = m n f k ∇ g k = ( f n g n − f m − 1 g m − 1 ) − ∑ k = m n g k − 1 ∇ f k , {\displaystyle \sum _{k=m}^{n}f_{k}\nabla g_{k}=\left(f_{n}g_{n}-f_{m-1}g_{m-1}\right)-\sum _{k=m}^{n}g_{k-1}\nabla f_{k},}

Both can be used to obtain a more symmetric version:

∑ k = m n ( f k Δ g k + g k ∇ f k ) = f n g n + 1 − f m − 1 g m {\displaystyle \sum _{k=m}^{n}\left(f_{k}\Delta g_{k}+g_{k}\nabla f_{k}\right)=f_{n}g_{n+1}-f_{m-1}g_{m}}

Summation by parts is an analogue to integration by parts:

∫ f d g = f g − ∫ g d f , {\displaystyle \int f\,dg=fg-\int g\,df,}

or to Abel's summation formula:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Summation by parts

Start with the simplest possible case. Write down what Summation by parts 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 Summation by parts 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 Summation by parts 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 Summation by parts

In research: Summation by parts 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 Summation by parts 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: Summation by parts is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lemmas in algebra, Real analysis, Summability methods, so understanding it makes those chapters shorter.
In everyday life: Look for Summation by parts 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 Summation by parts in 20 minutes

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

Frequently asked questions

What is Summation by parts in simple terms?

Why does Summation by parts 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 Summation by parts?

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 Summation by parts.