Sigma notation ninja tricks 1: jumping constants

Almost exactly ten years ago, I wrote a page on this blog explaining big-sigma notation. Since then it’s consistently been one of the highest-traffic posts on my blog, and still gets occasional comments and questions. A few days ago, a commenter named Kevin asked,

Could you explain how to take a constant outside of a summation and bring it inside the summation?

This made me realize there’s a lot more still to be explained! In particular, understanding what sigma notation means is one thing, but becoming fluent in its use requires learning a number of “tricks”. Of course, as always, they’re not really “tricks” at all: understanding what the notation means is the necessary foundation for understanding why the tricks work!

Trick 1: jumping constants

For today, we’ll start by considering what Kevin asked about. Consider what is meant by this sigma notation:

$\displaystyle \sum_{i=1}^{4} c X_i$

It doesn’t really matter what the $X$ ’s are, the point is just that each $X_i$ might be different, whereas $c$ is a constant that doesn’t change. So this can be expanded as

$\displaystyle \sum_{i=1}^{4} c X_i = c X_1 + c X_2 + c X_3 + c X_4$

Since multiplication distributes over addition, we can factor out the $c$ :

$c X_1 + c X_2 + c X_3 + c X_4 = c (X_1 + X_2 + X_3 + X_4)$

The right-hand side can now be written as

$\displaystyle c \left( \sum_{i=1}^4 X_i \right),$

so overall we have shown that

$\displaystyle \sum_{i=1}^4 c X_i = c \left(\sum_{i=1}^4 X_i\right).$

We usually omit the parentheses and just write

$\displaystyle c \sum_{i=1}^4 X_i.$

Our argument didn’t really depend on any of the specifics (like the fact that $i$ goes from $1$ to $4$ ). The general principle is that constants can “jump” back and forth across the sigma, which corresponds to multiplication distributing across addition.

The one remaining question is—what counts as a “constant”? The answer is, anything that doesn’t depend on the index variable. So the “constant” can even involve some variables, as long as they are other variables! For example,

$\displaystyle \sum_{i = 1}^k (n^2 + k) g(i) = (n^2 + k) \sum_{i=1}^k g(i)$

In the context of this sum, $n^2 + k$ is a “constant”, because it does not have $i$ in it. Since it doesn’t contain $i$ , it is going to be exactly the same for each term of the sum, which means it can be factored out.

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

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