🧮Series & Sigma Notation

A sequence is a list of numbers. A series is that list added up. Slope Glacier is measured in both: the depth of each yearly ice layer is a sequence, and the depth of the whole glacier is a series.

The story every teacher tells is true. A boy called Gauss was asked to add the whole numbers from 1 to 100 and had an answer almost at once. He did not add 100 numbers. He folded the list in half:

1 + 2 + 3 + ... + 98 + 99 + 100

The first and last add to 101. So do the second and second-last: 2 + 99 = 101. So do 3 and 98. Every pair adds to the same total, because as you walk in from one end you gain 1 and from the other end you lose 1. There are 100 numbers, so there are 50 pairs, and the sum is 50 * 101 = 5050.

That fold works for any arithmetic series — any list where each term is the one before plus a fixed common difference d. Write it as a formula and you have the whole of the first half of this craft:

S_n = n(a_1 + a_n)/2

In words: the number of terms, times the average of the two ends. Ren says it the second way when a student forgets the halving: you are not adding the ends, you are finding what an average term is worth and then paying it n times over.

There is a second form for when the last term is not given to you. Since a_n = a_1 + (n - 1)d, substituting gives

S_n = n(2a_1 + (n - 1)d)/2

which is the same formula wearing different clothes. Use whichever one matches what the question hands you.

A geometric series multiplies instead of adding: each term is the one before times a fixed common ratio r. Pairing the ends is no help here, because the terms are nowhere near evenly spaced. A different trick does the work.

Write the sum out, then write r times the sum underneath it:

S = a + ar + ar² + ... + ar^n-1

rS = ar + ar² + ... + ar^n-1 + arⁿ

Almost every term appears in both lines. Subtracting removes all of them at once:

S - rS = a - arⁿ

S(1 - r) = a(1 - rⁿ)

S = a_1(1 - rⁿ)/(1 - r)

That is the finite geometric sum, and the derivation is worth doing once by hand, because it explains the two things students get wrong. The exponent is n, the number of terms — not n - 1, which is the exponent in the *last term*. And the two brackets must be written the same way round: (1 - rⁿ) over (1 - r). If you prefer both to be positive, flip both at once and use a_1(rⁿ - 1)/(r - 1). The two forms give exactly the same number; flipping only one gives the answer with the wrong sign.

Try it on 3 + 6 + 12 + 24 + 48. Here a_1 = 3, r = 2, n = 5, so

S = 3(1 - 2⁵)/(1 - 2) = 3(1 - 32)/(-1) = 3(-31)/(-1) = 93

and adding the five terms by hand gives 93 too.

Mathematicians write "add these up" with the Greek capital sigma, Σ. It is shorthand, nothing more.

Σ_(k=1)^(8) (3k + 2)

is read "the sum from k = 1 to 8 of (3k + 2)", and it is an instruction: put k = 1 into the rule, then k = 2, and keep going up to k = 8, adding every result. Three parts carry all the meaning.

• ◆The rule in the brackets says what a term looks like.
• ◆The lower limit says where k starts, almost always 1.
• ◆The upper limit says where k stops. It is the number of terms, not the last term's value.

Reading one is mechanical. To evaluate Σ_(k=1)^(8) (3k + 2), split it: the 3k parts give 3(1 + 2 + ... + 8) = 3 * 36 = 108, and the + 2 sits inside every term, so it is added 8 times over for another 16. The total is 124. Three shortcuts are worth memorising for this:

1 + 2 + ... + n = n(n + 1)/2

1² + 2² + ... + n² = n(n + 1)(2n + 1)/6

c added n times = cn

Writing one is the reverse, and it is two questions. How many terms are there? That fixes the upper limit. What rule sends k = 1 to the first term? That fixes the brackets. For 2 + 5 + 8 + ... + 29 the common difference is 3, so the rule looks like 3k + something; at k = 1 it must give 2, so the something is -1 and the rule is 3k - 1. The number of terms is (29 - 2)/3 + 1 = 10. So the series is Σ_(k=1)^(10) (3k - 1), and the 29 never appears anywhere in the sigma.

For a geometric series the rule is a_1 r^k-1, and that k - 1 matters: at k = 1 it gives r⁰ = 1, so the first term comes out as a_1 exactly.

Add a geometric series for ever and one of two things happens.

If the size of r is 1 or more, each term is at least as big as the one before. The running total climbs and climbs and never settles, so the series has no sum at all. It does not converge.

If the size of r is less than 1, each term is a shrinking slice of what is left. 8 + 4 + 2 + 1 + 0.5 + ... never passes 16, and it gets as close to 16 as you like. That series converges, and its sum comes straight from the finite formula: as n grows, rⁿ shrinks towards 0, so a_1(1 - rⁿ)/(1 - r) settles on

S = a_1/(1 - r), valid only when the size of r is less than 1

The condition is not decoration. If you feed r = 3 and a_1 = 6 into the formula you get 6/(1 - 3) = -3, which is nonsense: every term of that series is positive and growing. Always check the ratio before you use the formula.

The prettiest use of this is turning a repeating decimal into a fraction. The decimal 0.727272... is really a series:

0.72 + 0.0072 + 0.000072 + ...

with a_1 = 72/100 and r = 1/100. So S = (72/100) / (99/100) = 72/99, and cancelling 9 from both parts gives 8/11. Check it on a calculator: 8 divided by 11 is 0.727272... The pattern generalises — the denominator gets one 9 for each digit in the repeating block — but the answer is only finished once it is in lowest terms.

Series turn up wherever something happens over and over with a steady pattern.

• ◆A savings plan. Put away 15 gold this month and 4 gold more each month than the last. After 12 months the amounts form an arithmetic series, and the total saved is 12(15 + 59)/2 = 444 gold.
• ◆Seats in an arena. The front row holds 22, and each row behind holds 3 more. Eighteen rows hold 18(22 + 73)/2 = 855 seats.
• ◆A message spreading. Each person tells 3 new people. The rounds are 3, 9, 27, ... — geometric with r = 3, and six rounds reach 3(3⁶ - 1)/2 = 1092 people.
• ◆A bouncing ball. Dropped from 12 m, rising to 2/3 of its height each time. The drop is 12 m once; the bounces are an infinite geometric series with a_1 = 8 and r = 2/3, summing to 24 m, and each bounce height is travelled twice, up and down. Total: 12 + 48 = 60 m.

How to type your answer here.

• ◆A sum or a total is typed as a plain number: 444. If it is not whole — which happens with r = 1/2 and with every repeating decimal — type a fraction in lowest terms, 45/8 or 8/11, and the prompt will say so. 90/16 is marked as not yet finished, not as wrong.
• ◆A count of terms or a missing first term or ratio is a whole number: 10.
• ◆Where the question asks you to write a series in sigma notation, or whether an infinite series converges, pick the option that matches — sigma symbols are not something you can type into the box, so those questions are multiple choice.
• ◆Nothing here needs a calculator's decimals. Keep fractions as fractions right to the end.