Intuition The one core ideaA never-ending sum of shrinking positive numbers is really just a staircase of rectangles , and the smooth curve those rectangles hug is something we already know how to measure: area . If we can measure the area, we can decide whether the staircase reaches the sky or stops at a finite height.
Before you can read the Integral Test note , you must own every symbol it throws at you. This page builds them in order , each from the picture it stands for. Nothing here assumes you have seen calculus notation before line one.
Definition How to read the::: self-test lines
Throughout this page you will see short lines shaped like Question ::: Answerflip-card prompts : read the part before the three colons, answer it in your head, then check yourself against the part after the three colons. The three colons are just the divider between question and hidden answer.
Definition Sequence and the terma n sequence is an ordered list of numbers, one for each counting number 1 , 2 , 3 , … n a n a-sub-n ". The little n slot number .
The picture: imagine numbered mailboxes in a row. Box 1 a 1 2 a 2
Intuition Why the topic needs thisThe whole subject is about adding these box-contents together. So we first need a clean name — a n n
a n For the list 1 , 2 1 , 3 1 , … a n = n 1 a 5 = 5 1
∑ ∑ "add these up" . Underneath it we write where the slot number starts, above it where it stops.
∑ n = 1 4 a n = a 1 + a 2 + a 3 + a 4 .
WHAT this says: start at n = 1 a n n = 4
Definition An infinite seriesWhen the top of the sigma is the sideways-eight ∞ infinity — "keeps going, no last number"), we mean we never stop adding:
∑ n = 1 ∞ a n = a 1 + a 2 + a 3 + ⋯ series .
∞ decide the fate of such an endless sum without doing the impossible addition. So the symbol ∞
∑ n = 1 3 n 1 1 + 2 1 + 3 1 = 6 11
S N stop at slot N and look at that finite total:
S N = ∑ n = 1 N a n = a 1 + a 2 + ⋯ + a N . N N
The picture: watch the total grow as you add one more box each time. S 1 , S 2 , S 3 , …
Definition Convergence of a sequence, written withlim S 1 , S 2 , S 3 , … L lim N → ∞ S N = L , S N and stay as close as we like to the single finite number L N S N L no such finite L diverges .
Intuition Why partial sums are the real definition"Does the infinite sum settle?" is literally the question "does lim N → ∞ S N L S N S N
n any convergence test, glance at the terms themselves. If a series ∑ a n must shrink to nothing :
lim n → ∞ a n = 0. lim n → ∞ a n = 0 diverges immediately — no further work needed. This is the n-th term test for divergence .
Intuition Necessary, not sufficientTerms going to zero is a doorway you must pass through, not a guarantee of arrival. Passing it (a n → 0 allowed to ask the deeper question the integral test answers. Failing it ends the story on the spot. The harmonic series has a n = n 1 → 0
Why is a n → 0 It is necessary but not sufficient; ∑ n 1 a n → 0
Definition Increasing and bounded above
A sequence is increasing if each term is ≥ S 1 ≤ S 2 ≤ S 3 ≤ ⋯
It is bounded above if there is a ceiling number M S N ≤ M N
The least upper bound (supremum) of the sequence is the smallest such ceiling — the lowest number L S N ≤ L N
If the a n positive , then adding a positive amount each step means the partial sums only ever go up — they are automatically increasing.
Intuition The monotone ceiling picturePicture the running totals climbing a ladder, never stepping down. A ceiling M lowest height nothing ever exceeds (the least upper bound). That "must settle" fact is exactly what the integral test uses in its final step: the integral supplies the ceiling M
Why "positive terms" matters With positive terms the totals only rise, so a ceiling forces convergence; with mixed signs totals can wobble and this reasoning breaks.
Definition Function and its curveA function f x f ( x ) graph is the smooth curve you get by plotting the point ( x , f ( x )) x is the output f ( x )
[ 1 , ∞ ) f domain [ 1 , ∞ ) x 1 [ 1 included ; the "∞ ) N [ N , ∞ ) [ 1 , ∞ )
The bridge to sequences: if we set a n = f ( n ) x = 1 , 2 , 3 , …
Intuition Why swap a sum for a functionA jagged sum is hard; a smooth curve has a tool we know — area . By turning a n f ( n )
Definition Positive, continuous, decreasing (on[ 1 , ∞ ) f [ 1 , ∞ )
Positive : the curve stays above the x f ( x ) > 0 Continuous : the curve has no breaks or jumps — so the area under it actually exists.Decreasing : the curve falls as you move right, so a later input never gives a taller output. This means on each width-1 left edge is the tallest point and the right edge is the shortest. Because every strip leans the same way, the rectangles can be compared with the area in one consistent direction — left-edge rectangles always sit above the curve, right-edge rectangles always below.
Definition The integral sign∫ ∫ a b f ( x ) d x exact area trapped between the curve y = f ( x ) x x = a x = b ∫ d x x
Intuition Why area answers our questionThe rectangles of the series and the area under the curve are so close they trap each other. If the smooth area is finite, the boxed rectangles can't total more than that finite ceiling — the series converges. If the area is infinite, the rectangles are dragged up with it. Area decides the fate of the sum. This is the heart of the whole method.
Definition Improper integral and the limitlim out to infinity we can't just plug in ∞ b b ∫ 1 ∞ f ( x ) d x = lim b → ∞ ∫ 1 b f ( x ) d x . lim b → ∞ b converges ; if it grows forever it diverges . See Improper integrals .
Why we need a limit here We can never "reach" infinity, so we track the area to a wall b
p n p 1 p power the slot number is raised to. A negative exponent just means "one over": x − p = x p 1 p faster , which is exactly the knob that decides convergence.
Definition The natural logarithmln ln x to what power must the special number e ≈ 2.718 x " It appears because the area under the curve y = 1/ x ln x
Meaning of x − 2 x 2 1 x
Why ln The area under y = 1/ x 1 b ln b ln b → ∞ ∑ n 1
The diagram below (rendered where Mermaid is available) shows the build order; if it appears as plain text, read it top-to-bottom as the same dependency chain described in §1–§8.
Monotone convergence theorem
Function f x and its graph
Positive continuous decreasing
Improper integral with limit
Test yourself — reveal only after answering aloud.
What does a n The number sitting in slot n
What does ∑ n = 1 ∞ a n Add every list-term forever, starting at slot 1.
What is the partial sum S N The finite running total a 1 + ⋯ + a N N
Write convergence of the series with limit notation. ∑ a n lim N → ∞ S N = L L L
State the n If ∑ a n lim n → ∞ a n = 0 a n → 0
State the Monotone Convergence Theorem. An increasing sequence bounded above converges, to its least upper bound (supremum).
Why does "increasing and bounded above" force convergence? Rising totals blocked by a ceiling must pile up under their least upper bound.
On what domain must f [ 1 , ∞ )
Name the three conditions on f Positive (no cancelling), continuous (area exists), decreasing (clean rectangle comparison).
What does ∫ 1 b f ( x ) d x The exact area between the curve and the x b
What does lim b → ∞ Slides the right wall to infinity and reports the number the area approaches.
What does x − p 1/ x p
What question does ln x To what power must e ≈ 2.718 x
Integral test — proof, p-series (index 4.3.7) — the topic these foundations unlock.Improper integrals — the limit-to-infinity area machine of §7.Harmonic series — where the slow ln n-th term test for divergence — the necessary (not sufficient) precondition of §4.Comparison test · Limit comparison test — sibling tests built on the same a n = f ( n ) Riemann zeta function — names the value of ∑ 1/ n p 
