Sequences, sums and series
Three pages of the formulary, explained formula by formula: what it is, why it exists, how to use it. Short sentences. A worked numerical example throughout. And tools to try the formulas yourself.
0Symbols
A formula isn't hard. It's a sentence written in shorthand. Here's the dictionary. Type in the filter box above to find a symbol.
| Symbol | Meaning |
|---|---|
| n | A rank. A position in the list. Always a whole number. |
| tₙ | The term at position n. The small number below isn't a multiplication: it's a place number. |
| t₁ | The very first term. The starting point. |
| tₙ₋₁ | The term right before the one at position n. The left neighbor. |
| r | The common difference or common ratio. The step of the progression: what you add, or what you multiply by. |
| Sₙ | The sum of the first n terms. |
| S∞ | The sum of every term, without ever stopping. |
| ∞ | Not a number. A direction: “it keeps going forever.” |
| Σ | A big Greek S. It says: add up everything that follows. The bottom shows where to start, the top where to stop. |
| k | A counter. It takes every value one after another. |
| uk | The term numbered k. Same idea as tₙ: the letter changes, the meaning doesn't. |
| sₙ | The partial sum: add up the first n terms and stop there. |
| lim | The value you're heading toward if you keep going long enough. You don't necessarily reach it. |
| → | “tends to,” “gets closer to.” |
| |r| | The absolute value: the number without its sign. |−3| equals 3. |
| rⁿ | r multiplied by itself n times. |
| ᵏ√uk | The kth root: what number, raised to the power k, gives back uk? |
| c | A name given to the result of a limit, so you can refer to it afterward. |
t₂ means “the second term.”
t² means “t multiplied by t.” The first sits low, the second sits high.1Progressions
A progression is a list of numbers built with a fixed rule. Always the same rule, from one term to the next. There are only two kinds.
Arithmetic: you always add the same thing. 3, 7, 11, 15, 19…
Geometric: you always multiply by the same thing. 3, 6, 12, 24, 48…
That's it. The whole chapter follows from these two sentences.
1.1 · The construction rule
It says how to build a term from the one before it.
“The term at position n equals the previous term, plus the common difference.”
It's the definition. It's the progression's DNA.
Everything else in the table is just a consequence of this one line.
You start from the first term. You apply the rule. You repeat.
Perfect for moving step by step. Bad for going far.
For the 500th term, you'd need 499 operations. Hence the next formula.
1.2 · The nth term
It's the shortcut. It hands you any term at once, without going through all the ones before it.
Because going step by step is too slow.
We want to jump straight to the right place.
Count the steps, not the terms.
From the 1st to the 5th term, you take 4 steps. Not 5.
That's the whole explanation for n − 1. There's nothing more to it.
Each step adds r → you add r a number of times equal to n − 1.
Each step multiplies by r → you multiply just as many times: that's rⁿ⁻¹.
t₁ + 0 × r = t₁ ✓ t₁ × r⁰ = t₁ ✓
With
n instead of n − 1, it doesn't come out right. That's how you spot the mistake.1.3 · Symmetry
Take a term a little after the start. Take the term equally far before the end.
Add them: you land back on the sum of the first and the last.
What you gain on one side, you lose on the other.
Moving two places forward adds the common difference twice. Moving two places back removes it twice.
The two movements cancel out exactly.
Almost never used as such in exercises.
It serves to prove the sum formula. That's its real job.
1.4 · The arithmetic sum
The number of terms, multiplied by the average of the first and the last.
Write out the list. Write it a second time underneath, backward.
Add column by column: each column gives the same total, t₁ + tₙ.
There are n columns → the total is n × (t₁ + tₙ).
But you counted the list twice. Divide by two.
1.5 · The geometric sum
Two ways of writing the same result.
The first if you know the first term and the number of terms.
The second if you know the last term.
Call the sum S. Multiply everything by r: it's the same list, shifted by one spot.
Subtract one from the other: the whole middle cancels out.
Only two terms remain, the first and the last. Hence the formula.
One precaution only: the common ratio must not equal 1.
Otherwise you'd divide by zero. And if r = 1, every term is equal: the sum is n × t₁.
1.6 · The infinite sum
You add up infinitely many terms. And you get a finite number.
If the ratio is smaller than 1 in size, the terms shrink very fast.
After a while, whatever you add changes nothing anymore.
Technically: in the formula from 1.5, rⁿ collapses toward zero. It disappears. What's left is t₁/(1 − r).
First check that |r| < 1. Always. That's the gateway.
If the condition isn't met, the sum doesn't exist. There's nothing to give.
2Power sums
Here the subject changes. We're no longer adding up a progression: we're adding up the integers, then their squares, then their cubes.
The small k above Sₙᵏ isn't a power. It's a label: “the version for the kth power.”
| Sum | Term-by-term addition | By the formula |
|---|
Integers. The rank, times its right-hand neighbor, divided by two. It's formula 1.4 applied to 1, 2, 3, 4…
Squares. Three factors: n, then n + 1, then 2n + 1. Divide by 6.
Cubes. It's the square of the sum of the integers. Always.
Fourth powers. The same three factors as the squares, plus a fourth, divided by 30.
These are no longer progressions: the gaps change at every step. No simple trick works.
But a regularity remains: the sum of the kth powers always gives a polynomial of degree k + 1.
Its leading term is nk+1/(k + 1), the area under the curve. These formulas are the exact version of an integral.
3Series
A sequence is a list. A series is what you get by adding that list all the way to infinity.
It isn't the same thing. It's the central distinction of the chapter.
3.1 · What “converging” means
We don't know how to add up infinitely many numbers. So we cheat.
We add the 1st. Then the first 2. Then the first 3. And so on.
We get a new list: the list of running totals.
Because this detour turns an impossible problem into a familiar one.
“What does this infinite sum equal?” becomes “what does this list of totals tend toward?”
And that, we know how to do: it's a limit.
The totals settle around a value → the series converges. That value is its sum.
The totals run off to infinity or oscillate without settling → the series diverges. There is no sum.
The lab above draws exactly this list of totals.
3.2 · The divergence test
The first reflex. The three-second test.
If the terms don't become tiny, you keep adding non-negligible amounts forever.
The total can't settle down. It's plain common sense.
This test only works in one direction.
Terms that don't tend to zero → diverges. Certain.
Terms that tend to zero → we know nothing. Keep looking.
The terms really do tend to zero. And yet the sum grows without bound. It's just very slow: it takes billions of terms to pass 20. But it never stops.
The sentence to remember: tending to zero is necessary, but not sufficient.
Try it in the lab above: the curve keeps climbing.
3.3 · Absolute convergence
Erase every minus sign. Make every term positive.
If it still converges under those conditions, it's solid.
Minus signs help convergence: they cancel out part of what you just added.
A series that converges even without that help really converges.
Very practical use: the tests that follow only accept positive terms.
Does your series have alternating signs? Switch to absolute values, then test.
If that converges: you've won, the original series converges too.
If that diverges: careful, you can't conclude anything about the original series.
3.4 · The ratio test (d'Alembert)
c < 1 → converges · c = 1 → uncertain · c > 1 → diverges
Compare each term to the one before it. Divide one by the other.
That division answers a simple question: by how much does the list shrink at each step?
Remember the infinite geometric sum: it converges if |r| < 1.
This test asks: does my series behave, in the long run, like a geometric one?
And if so, with what ratio? That ratio is c.
You then apply the rule for geometric progressions. The same idea, recycled.
3.5 · The root test (Cauchy)
c < 1 → converges · c = 1 → uncertain · c > 1 → diverges
Same conclusion, different route.
We ask: if my term were a kth power, what would its base be?
Factorials? Take the ratio: they simplify beautifully.
A large kth power? Take the root: it cancels it in one stroke.
Good to know: whenever the ratio test settles it, the root test settles it too, with the same value.
The reverse isn't true. The root is the stronger of the two tools.
Remember this result: for 1/ka, it converges if a > 1, and diverges if a ≤ 1.
4The method
You're given a series. What to do, and in what order.
5Six common mistakes
- Confusing the index with the exponent. Low down, a place number. Up high, a repeated multiplication.
- Writing n instead of n − 1. You count the steps, not the terms. Always test at n = 1.
- Believing that “the terms tend to zero” is enough. The harmonic series is there to remind you otherwise.
- Confusing the sequence and the series. The sequence 1, ½, ¼… tends to 0. The series 1 + ½ + ¼ + … tends to 2.
- Applying the geometric sum with r = 1. You'd divide by zero. In that case, the sum is n × t₁.
- Drawing a conclusion when c = 1. The test said nothing. Roughly one series in two falls into this case.
6Test yourself
Six questions. Immediate answer, with the explanation.
7The vocabulary in French and German
This formulary comes from Swiss gymnasium (upper-secondary) classrooms, taught in French in some cantons and German in others. Here are the key terms in their original languages, useful whenever you work from a French or German-language source.
| Français | Deutsch | Note |
|---|---|---|
| une suite | eine Folge | The list itself. |
| une série | eine Reihe | The sum of the list. Don't confuse the two. |
| un terme | ein Glied | An element of the list. |
| le rang | die Stelle · der Index | The position. |
| la raison (arithm.) | die Differenz | German distinguishes the two cases; French says “raison” for both. |
| la raison (géom.) | der Quotient | · |
| la somme partielle | die Teilsumme | The total after n terms. |
| la limite | der Grenzwert | The value being aimed at. |
| converger / diverger | konvergieren / divergieren | · |
| la valeur absolue | der Betrag | The number without its sign. |
| la racine k-ième | die k-te Wurzel | · |
| la puissance | die Potenz | False friend: Potenz, never Macht. |
| la factorielle | die Fakultät | · |
| croissant / décroissant | steigend / fallend | · |
| la série harmonique | die harmonische Reihe | The counterexample worth knowing. |
| nécessaire / suffisant | notwendig / hinreichend | The decisive pair of the chapter. |