Formula sheet

Sequences and series — Formula Sheet

7 formulas from Calculus II, each with what it means, when to use it and the trap it sets.

n \to \infty lim a_{n} \neq = 0 ⟹ \sum a_{n} diverges

What it says: If the terms do not shrink to zero, the sum cannot settle.
When to reach for it: Always run it first — it is one line and it ends a third of series questions.
Watch out: It only ever proves divergence. Terms going to zero proves nothing: the harmonic series has terms going to zero and still diverges.

\sum \frac{n}{n + 1} diverges since a_{n} \to 1

No further test needed once the terms fail to vanish.

n = 0 \sum \infty a r^{n} = \frac{a}{1 - r} for ∣ r ∣ < 1

What it says: Each term is a fixed multiple of the one before, and if that multiple is small enough the total is finite.
When to reach for it: A constant ratio between consecutive terms — one of the few series with an exact sum.
Watch out: a is the first term as actually written. If the sum starts at n = 1, the first term is ar, not a.

n = 0 \sum \infty \frac{1}{2 ^{n}} = \frac{1}{1 - 1/2} = 2

a = 1 and r = 1/2, so the total is exactly 2.

n = 1 \sum \infty \frac{1}{n ^{p}} converges ⟺ p > 1

What it says: One exponent decides the whole question.
When to reach for it: The benchmark you compare almost every other positive series against.
Watch out: p = 1 is the harmonic series, and it diverges — the boundary case falls on the wrong side.

\sum \frac{1}{n ^{2}} converges, \sum \frac{1}{n} diverges

p = 2 and p = 1/2 respectively.

L = n \to \infty lim \frac{a _{n + 1}}{a _{n}}; L < 1 \Rightarrow converges

What it says: Compare each term with the one before; if the ratio settles below 1, the sum converges absolutely.
When to reach for it: Factorials, or a variable in an exponent — anything where the ratio simplifies dramatically.
Watch out: L = 1 is a genuine no-answer. Switch to comparison or the integral test instead of guessing.

\sum \frac{2 ^{n}}{n !} : L = lim \frac{2}{n + 1} = 0 < 1

Factorials always beat exponentials, so this converges.

\sum (- 1)^{n} b_{n} converges if b_{n} ↓ 0

What it says: Alternating signs with terms that decrease steadily to zero always converge.
When to reach for it: A (−1)ⁿ or (−1)^{n+1} out front, once the divergence test has come back clean.
Watch out: Convergence here can be conditional. Check Σ|aₙ| separately if the question asks about absolute convergence.

\sum \frac{( - 1 ) ^{n}}{n} converges, \sum \frac{1}{n} does not

The textbook example of conditional convergence.

\frac{a _{n + 1}}{a _{n}} ∣ x - c ∣ < 1 \Rightarrow R

What it says: The ratio test, applied with x still in it, gives the interval where the series makes sense.
When to reach for it: Any power series, before you are allowed to use it for anything.
Watch out: The two endpoints must be tested separately by hand — the ratio test says nothing at |x−c| = R.

\sum \frac{x ^{n}}{n} : R = 1, converges on [- 1, 1)

Convergent at −1, divergent at +1 — the endpoints really do differ.

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( c )}{n !} (x - c)^{n}

What it says: Rebuild a function from its derivatives at a single point.
When to reach for it: Approximating a function near a point, or turning an unintegrable expression into a polynomial.
Watch out: The n! is not optional and it is a factorial, not a product of the derivatives. c = 0 gives the Maclaurin series.

e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots

Worth memorising alongside sin x, cos x and 1/(1−x).

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