Summation Notation

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In previous work on series we have seen $S_n$ notation which is the sum of the first $n$ terms of a sequence,

$$S_n=U_1+U_2+U_3+...+U_n$$ $$\underset{i=1}{\overset{n}{\Sigma }}{x}_i=x_1+x_2+x_3+\mathrm{...}+x_n$$

Summation notation is similar to the $S$ notation used before, but it has greater flexibility describing the sum and has some similarities to integration. Summation notation is also called sigma (Σ) notation.

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The series $S_n$ can be written in sigma notation as

$$S_n=\underset{i=1}{\overset{n}{\Sigma }}{x}_i$$

• where
• $n\in \mathbb{N}$;
• and
• $x_i$ is an element of some given sequence

For the sum to infinity

$$S_{\infty }=\underset{i=1}{\overset{\infty }{\Sigma }}{x}_i$$

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Summation Notation and the Binomial Expansion

To demonstrate the flexiblility of summation notation here is the binomial expansion:

$${(a+b)}^n=\underset{i=0}{\overset{n}{\Sigma }}\left(\begin{array}{c}n\\ i\end{array}\right)a^{n-i}{b}^i$$

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Further Reading

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