Series — partial sums, convergence definition: Foundations

Before you can read the parent note on partial sums and convergence, you must own every symbol it fires at you. This page lists them in the order they build on each other — nothing appears before it is defined, and each thing is anchored to a picture.

1. The subscript $a_{n}$ — a numbered slot

The picture: imagine a row of numbered mailboxes. Box $1$ holds a number, box $2$ holds a number, and so on forever. $a_{n}$ is "whatever number lives in box $n$ ."

Figure — Series — partial sums, convergence definition

Why the topic needs it. A series adds up a specific list of numbers. To talk about "the general term" without writing them all out, we need one symbol, $a_{n}$ , that stands for the entry at any address $n$ .

2. A sequence $(a_{n})$ — the whole numbered row

The picture: the same row of mailboxes, but now you step back and look at the whole row as a single thing rather than one box at a time. The row has a shape — maybe the numbers shrink toward $0$ , maybe they grow, maybe they wobble.

Why the topic needs it. The parent note's punchline is "a series IS a sequence in disguise." You cannot understand that until sequence means "one object made of infinitely many ordered numbers." See Sequences — limits and convergence for the full story.

3. The limit $lim_{n \to \infty} a_{n} = L$ — where the row is headed

Why " $\to \infty$ " here, and not some finite stop? Because we are asking about the long-run behaviour. No finite address is special; the question is where the numbers settle if you never stop walking. The arrow $\to$ reads "approaches."

Why the topic needs it. Convergence of a series is defined as a limit — but a limit of the running totals, not of the terms. Keeping these two straight is the whole game (Mistake 2 in the parent note).

4. The summation sign $\sum$ — "add these up"

$\sum_{n = 1}^{4} a_{n} = a_{1} + a_{2} + a_{3} + a_{4} .$

The picture: a machine with a start dial ( $n = 1$ ) and a stop dial ( $N = 4$ ). It grabs one mailbox at a time, from start to stop, and pours all their numbers into one running bucket.

Why the top number matters. If the stop dial reads a finite $N$ , you get a genuine, finite sum — you can do it. If the top reads $\infty$ , you cannot literally add forever; that is exactly the problem the topic solves by watching partial sums instead.

5. The partial sum $s_{N}$ — a running total that stops

The picture: stand at your bucket. $s_{1}$ = pour in box 1. $s_{2}$ = pour in box 2 as well. $s_{3}$ = box 3 too. The bucket level after each pour is a partial sum. Watching the level rise pour-by-pour is watching the sequence $(s_{N})$ .

Why the topic needs it — the central trick. "Add infinitely many numbers" is undefined (addition only ever combines two things at a time). But $s_{1}, s_{2}, s_{3}, \dots$ is a perfectly ordinary sequence, and a sequence either has a limit or it doesn't. So the topic redefines the infinite sum as $\sum_{n = 1}^{\infty} a_{n} = lim_{N \to \infty} s_{N} .$ The impossible "add forever" becomes the familiar "does this sequence converge?"

6. Converge / diverge — does the bucket level settle?

The picture, both cases:

Converge: the bucket level rises but slows, hugging a fixed line $S$ it never quite crosses.
Diverge: the level either keeps rising with no ceiling, or it jumps up-down-up-down and never picks a home.

7. The quantifiers $\forall$ , $\exists$ and $ε$ — the precise "settles"

Putting them together, $lim_{N \to \infty} s_{N} = S$ means: $\forall ε > 0, \exists M s.t. N \geq M ⟹ ∣ s_{N} - S ∣ < ε .$

Plain words: pick any tolerance band of half-width $ε$ around $S$ (however thin). There is a cutoff $M$ after which every running total lands inside that band and stays.

Why we need this much precision. "Gets closer and closer" is vague — closer how fast, closer to what? The $ε$ -band makes "settles" testable: no matter how thin a band a skeptic draws, you can name a point past which you never leave it.

How these feed the topic

Read it upward: a single term $a_{n}$ builds a sequence; adding a finite range with $\sum$ builds a partial sum $s_{N}$ ; the partial sums form their own sequence; asking whether that sequence has a limit — made precise by $ε$ — is exactly what it means for the series to converge.

Where each foundation reappears

Foundation	First used in topic for...
$a_{n}$ , sequence	writing the general term of Geometric series
limit of $(a_{n})$	the n-th Term Test for Divergence ( $a_{n} \to 0$ ?)
$\sum$ , $s_{N}$	deriving every closed-form partial sum
telescoping cancellation	Telescoping series
$a_{n} \to 0$ but $\sum$ diverges	the Harmonic series

Equipment checklist

Test yourself — cover the right side.

What does

a_{3}

mean, and is the

3

a multiplier?

The third term of the list; the

3

is an address/label, not multiplication.

What is a sequence

(a_{n})

(versus a single term)?

The entire ordered infinite list

a_{1}, a_{2}, a_{3}, \dots

taken as one object.

What does

lim_{n \to \infty} a_{n} = L

say in plain words?

As addresses grow, the terms get and stay arbitrarily close to

L

Read

\sum_{n = 1}^{4} a_{n}

out loud and expand it.

"Sum of

a_{n}

from

n = 1

4

= a_{1} + a_{2} + a_{3} + a_{4}

Why can't

\sum_{n = 1}^{\infty} a_{n}

be computed by direct addition?

Addition only ever combines two numbers at a time; you can't perform infinitely many steps.

Define the partial sum

s_{N}

s_{N} = a_{1} + a_{2} + \dots + a_{N} = \sum_{n = 1}^{N} a_{n}

What sequence do we actually watch to decide convergence?

The sequence of partial sums

(s_{N})

, not the sequence of terms

(a_{n})

\sum a_{n}

converges to

S

means exactly what?

lim_{N \to \infty} s_{N} = S

, a finite number.

Does "diverges" always mean "goes to infinity"?

No — it means no finite limit; it can also bounce forever, like

- 1, 0, - 1, 0, \dots

What do

\forall

and

\exists

mean?

"For every" and "there exists."

What does

∣ s_{N} - S ∣

measure?

The distance between the running total and the target

S

(sign ignored).

Connections

Parent topic ↑
Sequences — limits and convergence (a series is a sequence of partial sums)
Geometric series · Telescoping series · Harmonic series
n-th Term Test for Divergence