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.

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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One Response to Sigma notation ninja tricks 1: jumping constants

  1. Pingback: Sigma notation ninja tricks 2: splitting sums | The Math Less Traveled

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