0.2.2 Sums and products: Σ and Π

What Σ means

The capital Greek letter $\Sigma$ (sigma) means "add up". Here is a sum written out in full and in sigma notation:

\begin{aligned} \sum_{i=1}^{4} i^2 &= 1^2 + 2^2 + 3^2 + 4^2 \\ &= 1 + 4 + 9 + 16 = 30 \end{aligned}

Read it as "the sum of $i^2$ for $i$ from 1 to 4". The letter $i$ is the index. It starts at the number below the $\Sigma$ , goes up by 1 each time, and stops at the number on top. The expression to the right of $\Sigma$ is the term: work it out for each $i$ and add the results.

Rule: Sigma notation

\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n

Start the index at $m$ , add one term for each whole number up to $n$ , and include $n$ itself. That makes $n - m + 1$ terms.

i = 4

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Sums of data

Most sums in this course add up a list of values. With $x_1 = 4$ , $x_2 = 8$ , $x_3 = 15$ , $x_4 = 16$ :

\begin{gathered} \sum_{i=1}^{4} x_i = 4 + 8 + 15 + 16 = 43 \\[6pt] \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{43}{4} = 10.75 \end{gathered}

The second formula is the average from lesson 0.2.1, written with $\Sigma$ . When the limits are the whole list, books often drop them and write $\sum_i x_i$ , or just $\sum x_i$ .

The name of the index does not matter. $\sum_{i=1}^{n} x_i$ and $\sum_{k=1}^{n} x_k$ are the same sum, just as renaming a loop variable does not change a loop.

\sum x_i

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Three rules for sums

Write a sum out in full and these rules are ordinary arithmetic. They let you pull numbers out of a sum and split it into simpler pieces.

Rule: Constant multiple

\sum_{i=1}^{n} c\, a_i = c \sum_{i=1}^{n} a_i

A number that multiplies every term can come out in front. Doubling every value doubles the total.

Rule: Sum of two sequences

\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i

A sum of sums can be split into two sums, and two sums can be joined into one.

Rule: Sum of a constant

\sum_{i=1}^{n} c = n\,c

Adding the same number $c$ once for each of $n$ terms gives $n$ times $c$ . For example, $\sum_{i=1}^{5} 3 = 15$ .

Together they handle sums like $\sum_{i=1}^{4} (2x_i + 1)$ with the four values above. Split it, pull out the 2, and add the constant four times: $2 \times 43 + 4 \times 1 = 90$ .

Try it Check the rules on the same data. Each line prints the long way and the short way.

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Weighted sums

A weighted sum multiplies each value by its own weight, then adds:

\sum_{i=1}^{n} w_i x_i = w_1 x_1 + w_2 x_2 + \cdots + w_n x_n

With weights $0.5, 0.3, 0.2$ and values $10, 20, 30$ , that is $5 + 6 + 6 = 17$ . When the weights add up to 1, as here, the result is a weighted average. Case study 0.2.4 uses exactly this sum for a portfolio.

Try it Pair up the weights and values with zip, or let NumPy do it. The @ operator multiplies in pairs and adds, which is a weighted sum in one symbol.

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Products with Π

The capital Greek letter $\Pi$ (pi) does for multiplication what $\Sigma$ does for addition:

\prod_{i=1}^{4} i = 1 \times 2 \times 3 \times 4 = 24

Rule: Pi notation

\begin{gathered} \prod_{i=m}^{n} a_i = a_m \cdot a_{m+1} \cdots a_n \\ \prod_{i=1}^{n} c = c^n \end{gathered}

Multiply the term for each index from $m$ to $n$ . Multiplying the same $c$ $n$ times gives $c^n$ .

Products appear whenever something grows in steps. Say a value grows by 10%, then falls by 5%, then grows by 20%. Write $g_i$ for the growth in step $i$ , so $g_1 = 0.10$ , $g_2 = -0.05$ and $g_3 = 0.20$ .

\begin{aligned} \prod_{i=1}^{3} (1 + g_i) &= 1.10 \times 0.95 \times 1.20 \\ &= 1.254 \end{aligned}

So the value ends up multiplied by 1.254. Lesson 1.4.3 shows how logarithms turn a $\Pi$ into a $\Sigma$ .

Try it math.prod is the product of a list, as sum is its sum. The second line prints 1.2539999999999998. That is rounding inside the computer, not an error.

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Empty sums and empty products

What if the top limit is below the bottom one, as in $\sum_{i=1}^{0} a_i$ ? Then there are no terms at all. By convention an empty sum is 0 and an empty product is 1.

These are the values that change nothing: adding 0 or multiplying by 1 leaves a total as it was. Python follows the same convention, and the coding exercise in lesson 0.2.3 relies on it.

Try it range(1, 1) is empty, so both of these have nothing to add or multiply.

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Common mistakes

\sum a_i^2 = 14

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Sums and products

0 of 9 correct

1.
Evaluate $\displaystyle\sum_{i=1}^{5} i$
2.
Evaluate $\displaystyle\sum_{k=0}^{3} 2^k$
3.
Evaluate $\displaystyle\sum_{i=1}^{6} 4$
4.
Evaluate $\displaystyle\sum_{i=2}^{5} (2i - 1)$
5.
How many terms does $\displaystyle\sum_{i=3}^{10} a_i$ have?
6.
If $\displaystyle\sum_{i=1}^{5} x_i = 20$ , find $\displaystyle\sum_{i=1}^{5} (3x_i - 2)$
7.
Weights 0.2 and 0.8, values 5 and 10. Evaluate $\displaystyle\sum_{i=1}^{2} w_i x_i$
8.
Evaluate $\displaystyle\prod_{i=1}^{5} i$
9.
Evaluate $\displaystyle\prod_{j=1}^{3} 2$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

\sum_{i=m}^{n} a_i

Sums and products: Σ and Π

What Σ means

Rule: Sigma notation

Sums of data

Three rules for sums

Rule: Constant multiple

Rule: Sum of two sequences

Rule: Sum of a constant

Weighted sums

Products with Π

Rule: Pi notation

Empty sums and empty products

Common mistakes

Sums and products

Summary card

Free further reading