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Sigma Notation and Series

Discrete Mathematics · Chapter 3: Sequences and SeriesSection 3.5 · lesson 21 of 8710 min · free with a free account

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In this lesson

A compact name for adding up a list - and rules for rearranging it safely.

  • Read and write summation (sigma) notation
  • Distinguish a sequence from its series
  • Prove and apply the linearity and splitting rules
  • Use partial sums as a running total

Transcript

Everything said in the video, section by section (1,528 words). Read along, or search it.

Sequences and Series

Sigma Notation and Series

This section introduces sigma notation, a compact shorthand for adding up a list of terms. We'll define a series as the sum a sequence produces, and prove the small set of rules that let a sum be split, recombined, and factored safely. And we'll meet partial sums, which are exactly a running total.

The Opening Problem

A Running Total, Every Morning

A small software company's finance team watches one number every morning, the running total of that month's cloud bill. Each night, a dashboard pulls the previous day's charge, say one hundred eighteen dollars and forty cents, and adds it to everything so far that month. It doesn't re-display the whole list for someone to re-add by hand. It needs a compact way to say, the sum of every daily charge from day one up through day k. And that description has to update cleanly every time k grows by one. Summation notation provides exactly that. And its rules let a billing system compute a total in whatever grouping is convenient, and still land on the correct figure.

Definition 3.5.1

The Anatomy of Sigma Notation

Let a sub i be a real number for every integer i from m to n. Then sigma, from i equals m to n, of a sub i, means a sub m plus a sub m plus one, and so on, up to a sub n. The sigma symbol signals repeated addition. i is the index variable. m, written below the sigma, is the lower bound. n, written above it, is the upper bound. And the expression to the right of the sigma is the summand, or general term. The index steps through every integer from the lower bound to the upper bound. At each step the summand is evaluated, and the results are added.

A Dummy Variable

Just Like a Loop Accumulator

That's the same structure a for-loop accumulator carries out, one iteration at a time. And the index is a bound, or dummy, variable. It exists only to describe the pattern of addition. Renaming it doesn't change the sum. Summing over i or over j gives the identical number. In the same way, renaming the loop variable from i to j doesn't change what a program computes.

Worked Example 3.5.1

A 5-Day Billing Window

Here's an illustrative five-day billing window. The charges are one hundred eighteen forty, ninety-five ten, one hundred forty twenty-five, one hundred ten seventy-five, and one hundred thirty dollars. The sum from i equals one to five of c sub i expands to those five terms, added. Adding them in order gives two hundred thirteen fifty, then three hundred fifty-three seventy-five, then four hundred sixty-four fifty, and finally five hundred ninety-four fifty. The five-day total is five hundred ninety-four dollars and fifty cents. Notice that the notation didn't change the arithmetic. It only gave the specific sum a compact name.

Definition 3.5.2

Sequence Versus Series

A sequence and the series built from it are related, but they're not the same object. Confusing them is a common early mistake. The series associated with a sequence is the sum of its terms. Every finite series has one well-defined value, by ordinary addition. An infinite series needs additional convergence machinery to assign it a value, which comes later. Take a week of app sign-ups. Forty-two, fifty-one, thirty-nine, sixty, forty-seven, fifty-five, fifty-eight. The sequence shows the dip on day three, and the peak on day four. The series is the single number three hundred fifty-two, the total. That's the headline figure, but only the sequence reveals which day needs investigating. The sequence and the series carry different information. That's why a dashboard usually shows both a day-by-day chart and a running total.

Theorem 3.5.1

Linearity of Summation

Summation obeys a small set of algebraic rules. The constant multiple rule says the sum of c times a sub i equals c times the sum of a sub i. A constant factors out. The sum rule says the sum of a sub i plus b sub i equals the sum of the a's, plus the sum of the b's. Together, they're called linearity of summation. Each one follows from the definition and the ordinary properties of addition. And each can be proved rigorously, by mathematical induction on the upper bound.

Proof by Induction

Why the Sum Rule Holds

Here's the idea of the proof for the sum rule. The base case is n equal to m. There's a single term, and both sides equal a sub m plus b sub m. Now assume the rule holds up to n equal to k. The sum through k plus one is the sum through k, plus the new term. Apply the assumption to the sum through k. Then regroup the four pieces, using commutativity and associativity of addition. That gives exactly the sum of the a's through k plus one, plus the sum of the b's. By induction, the rule holds for every n. The constant multiple rule follows the same pattern, finishing with the distributive law. Induction itself is developed fully in chapter four.

Proposition 3.5.2

The Splitting Rule

A third rule lets a sum be broken at any interior boundary. Take integers with m at most p, and p at most n. Then the sum from m to n equals the sum from m to p, plus the sum from p plus one to n. Stating it precisely needs one convention. A sum over an empty range, where the lower bound exceeds the upper bound, equals zero. That keeps the algebra consistent exactly at the edge. When p equals n, the second sum runs from n plus one to n, which is empty, and contributes zero.

Table 3.5.1

The Rules at Work

Here's what each rule is for. The constant multiple rule lets you factor out a fixed conversion rate, tax rate, or unit price, before summing or after. Either order gives the same total. The sum rule lets you combine two categories tracked separately, such as cost and a flat service fee, into one running total. The splitting rule lets you break a range at a natural boundary, like a plan change or a billing sub-period, or reconcile two partial reports against one total. And the empty-sum convention handles a range with no values, such as zero elapsed days.

Worked Example 3.5.3

Fees, Conversion, and a Check

Return to the five days of charges. Suppose each day also carries a flat five-dollar platform fee, and finance needs the total at an illustrative conversion factor of zero point nine two. By the sum rule, combine the two categories. The charges total five hundred ninety-four fifty, and the fees are five dollars five times, which is twenty-five. So the total is six hundred nineteen fifty. By the constant multiple rule, converting that total once gives five hundred sixty-nine ninety-four. That's guaranteed to match converting all ten amounts first, so finance can pick the cheaper order. Now check with the splitting rule, at day three. Days one to three contribute three hundred sixty-eight seventy-five. Days four and five contribute two hundred fifty seventy-five. Together, that's six hundred nineteen fifty, matching exactly. Grouping the days differently couldn't change the total.

Definition 3.5.3

Partial Sums: The Running Total

The dashboard's running total is a new mathematical object in its own right. The k-th partial sum, S sub k, is the sum of the first k terms, for k equal to one, two, three, and so on. The list of partial sums is itself a sequence. And when the original sequence has a last index n, S sub n equals the whole series. Computing partial sums is exactly what an accumulator variable does in a loop. It's updated once per term, adding the new term to the previous running total. It never recomputes the whole sum from scratch.

Figure 3.5.2

Charges and Their Running Total

This figure shows the five daily charges on the left, and their partial sums on the right. Each partial sum is the previous one plus that day's charge. One hundred eighteen forty, then two hundred thirteen fifty, then three hundred fifty-three seventy-five, then four hundred sixty-four fifty. And the last, five hundred ninety-four fifty, is the whole series. It's the running total a dashboard would display.

Worked Example 3.5.4

Recovering a Day From 2 Totals

Suppose the dashboard only ever displays running totals. A viewer sees S three, three hundred fifty-three seventy-five, and S four, four hundred sixty-four fifty, but never the day-four charge. Since S four equals S three plus c four, the day-four charge comes back by subtraction. Four hundred sixty-four fifty minus three hundred fifty-three seventy-five is one hundred ten seventy-five. That matches the original charge exactly. In general, each term equals S sub k minus S sub k minus one. So a dashboard needn't keep every day's charge on display. Any single day can be recovered by differencing two consecutive running totals.

Takeaways

Adding, Compactly and Safely

To summarize. Sigma notation is shorthand for repeated addition, built from an index variable, two bounds, and a summand. A series is the single number a sequence produces. It discards the order and individual detail the sequence keeps. Summation is linear. Constants factor out, and termwise sums combine, and both are proved by induction. A sum can be split at any interior point without changing its value. And consecutive partial sums always recover the original term. Here's something to think about. Does summation distribute over a termwise product? Is the sum of a i times b i equal to the product of the two sums? Test it on a small example. If it fails, what makes addition and constant multiplication work, when a product of two sequences doesn't?

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FAQ

Questions students ask us

About Sigma Notation and Series

What does sigma notation mean, and what are its parts?
Sigma notation \sum_{i=m}^{n} a_i is shorthand for the sum a_m + a_{m+1} + \cdots + a_n. The index i steps through every integer from the lower bound m to the upper bound n, and the summand a_i is evaluated and added at each step. Because i is a dummy variable, renaming it to j does not change the sum.
What is the difference between a sequence and a series?
A sequence is an ordered list of terms, while the series is the sum of those terms. For example, the sign-ups 42, 51, 39, 60, 47, 55, 58 form a sequence, and its series is the single number 352. The sequence shows which day dipped or peaked, but the series gives only the total, so the two carry different information.
What are the linearity and splitting rules for summation?
Linearity has two parts: \sum_{i=m}^{n} c\,a_i = c\sum_{i=m}^{n} a_i and \sum_{i=m}^{n} (a_i+b_i)=\sum_{i=m}^{n} a_i+\sum_{i=m}^{n} b_i. Splitting says that for m\le p\le n, \sum_{i=m}^{n} a_i=\sum_{i=m}^{p} a_i+\sum_{i=p+1}^{n} a_i. Do not extend these to products: \sum a_ib_i is generally not (\sum a_i)(\sum b_i).
What is a partial sum, and how do you recover a term from partial sums?
The k-th partial sum S_k=\sum_{i=1}^{k} a_i is the running total of the first k terms. Each one is the previous total plus the new term, S_k=S_{k-1}+a_k, so a_k=S_k-S_{k-1}. For example, if S_3=353.75 and S_4=464.50, then a_4=464.50-353.75=110.75.

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