What does this Calculus II: Sequences & Series cheat sheet cover?

It has 11 sections: Sequences, Series Basics, p-Series, Divergence & Comparison, Integral Test, Ratio & Root, Alternating Series, Absolute Convergence, Power Series, Taylor & Remainder and Remainder. Together they cover the topics a typical Calculus II: Sequences & Series exam tests, including geometric series, p-series, divergence test and integral test.

Is the series convergence tests cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Calculus II: Sequences & Series cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Calculus II: Sequences & Series sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathCalculus II: Sequences & Series

Series Convergence Tests Cheat Sheet — 1-page PDF

Convergence tests, power series, Taylor/Maclaurin, and remainder estimates. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Calculus II Final exam study sheet.

1 pageUS Letter
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Preview of the one-page Calculus II: Sequences & Series cheat sheet: Convergence tests, power series, Taylor/Maclaurin, and remainder estimates — the whole course, one page

On the sheet

What’s on it

The topics a typical Calculus II: Sequences & Series exam tests, all on the one page.

Geometric series
P-series
Divergence test
Integral test
Comparison and limit comparison
Ratio and root tests
Alternating series test and error bound
Absolute vs conditional convergence
Power series radius/interval of convergence
Taylor and Maclaurin series table
Taylor remainder

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The free part, as text

The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Sequences

Sequence: An ordered list ${a_{n}}$ ; limit $lim_{n \to \infty} a_{n} = L$ iff $a_{n}$ approaches $L$ .

Sequence limit

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

If $L$ exists, sequence converges; else diverges.

Divergence to infinity

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

Sequence grows without bound; not convergent.

Common sequence facts

Form	Limit
$\frac{1}{n ^{p}}$ , $p > 0$	$0$
$r^{n}$ , $∣ r ∣ < 1$	$0$
$r^{n}$ , $∣ r ∣ > 1$	diverges
$\frac{n ^{k}}{a ^{n}}$ , $a > 1$	$0$

Watch out:For sequences, $lim a_{n}$ is a number or $\pm \infty$ ; for series, test $lim a_{n}$ with the nth-term test on terms $a_{n}$ of $\sum a_{n}$ .

02Series Basics

Infinite series: $\sum_{n = 1}^{\infty} a_{n}$ ; partial sums $s_{N} = \sum_{n = 1}^{N} a_{n}$ . Series converges iff $lim_{N \to \infty} s_{N}$ exists.

03Divergence & Comparison

Limit comparison

lim_{n \to \infty} \frac{a _{n}}{b _{n}} = c

If $0 < c < \infty$ and $a_{n}, b_{n} > 0$ , then $\sum a_{n}$ and $\sum b_{n}$ both converge or both diverge.

Nth-term test

lim_{n \to \infty} a_{n} \neq = 0 \Rightarrow \sum a_{n} diverges

Only a divergence test; $lim a_{n} = 0$ is inconclusive.

Choosing a comparison

Looks like	Compare to
$\frac{1}{n ^{p} + 5}$	$1/ n^{p}$
$\frac{n}{n ^{2} + 1}$	$1/ n$
$\frac{3 n ^{2} + 1}{n ^{4} - 7}$	$1/ n^{2}$

LCTest $\sum \frac{2 n + 1}{n ^{2} + 3}$ .

For large $n$ , like $1/ n$ . Limit comparison with $b_{n} = 1/ n$ : $lim \frac{( 2 n + 1 ) / ( n ^{2} + 3 )}{1/ n} = lim \frac{2 n ^{2} + n}{n ^{2} + 3} = 2$ . Since $\sum 1/ n$ diverges, series diverges.

04Integral Test

Integral test

$a_{n} = f (n), f$ positive, continuous, decreasing on $[N, \infty)$

Then $\sum_{n = N}^{\infty} a_{n}$ converges iff $\int_{N}^{\infty} f (x) d x$ converges.

05Alternating Series

Alternating series: Series $\sum (- 1)^{n} b_{n}$ or $\sum (- 1)^{n - 1} b_{n}$ with $b_{n} \geq 0$ .

Alternating series test

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

Need $b_{n}$ decreasing to $0$ .

Alternating error bound

∣ R_{N} ∣ = ∣ S - S_{N} ∣ \leq b_{N + 1}

Approximation error after $N$ terms is at most next omitted term.

ASTDoes $\sum (- 1)^{n - 1} \frac{1}{n}$ converge?

$b_{n} = 1/ n ↓ 0$ ⇒ converges by AST, but not absolutely since $\sum 1/ n$ diverges ⇒ conditional convergence.

ErrorApproximate $\sum_{n = 1}^{\infty} (- 1)^{n - 1} / n$ using first 4 terms.

$∣ R_{4} ∣ \leq b_{5} = 1/5$ .

06Absolute Convergence

Absolute convergence: $\sum a_{n}$ converges absolutely iff $\sum ∣ a_{n} ∣$ converges.
Conditional convergence: $\sum a_{n}$ converges but $\sum ∣ a_{n} ∣$ diverges.

Absolute ⇒ convergence

\sum ∣ a_{n} ∣ converges \Rightarrow \sum a_{n} converges

Then convergence is absolute.

07Taylor & Remainder

Taylor series

f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

If the series equals $f$ on its interval of convergence.

Maclaurin

a = 0

Taylor series centered at $0$ .

Standard Maclaurin series

$f (x)$	Series
$e^{x}$	$\sum_{n = 0}^{\infty} \frac{x ^{n}}{n !}$
$sin x$	$\sum_{n = 0}^{\infty} (- 1)^{n} \frac{x ^{2 n + 1}}{( 2 n + 1 )!}$
$cos x$	$\sum_{n = 0}^{\infty} (- 1)^{n} \frac{x ^{2 n}}{( 2 n )!}$
$\frac{1}{1 - x}$	$\sum_{n = 0}^{\infty} x^{n}$ , $∣ x ∣ < 1$
$ln (1 + x)$	$\sum_{n = 1}^{\infty} (- 1)^{n - 1} \frac{x ^{n}}{n}$ , $- 1 < x \leq 1$

More standard series

$f (x)$	Series / condition
$arctan x$	$\sum_{n = 0}^{\infty} (- 1)^{n} \frac{x ^{2 n + 1}}{2 n + 1}$ , $∣ x ∣ \leq 1$
$(1 + x)^{k}$	$\sum_{n = 0}^{\infty} (n k) x^{n}$ , $∣ x ∣ < 1$
$\frac{1}{1 + x ^{2}}$	$\sum_{n = 0}^{\infty} (- 1)^{n} x^{2 n}$ , $∣ x ∣ < 1$

TaylorMaclaurin for $cos x$ up to $x^{4}$ .

$cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} + \dots$

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Also on the full Calculus II: Sequences & Series sheet

8 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Series Basics

Necessary condition
Geometric series
Finite geometric sum

p-Series

p-series
Harmonic series
Quick compare

Divergence & Comparison

Direct comparison

Integral Test

Integral bounds

Ratio & Root

Ratio test
Root test
Best use

Absolute Convergence

Decide type

Power Series

Power series
Radius / interval
Ratio-test radius
Root-test radius

Remainder

Lagrange remainder
Lagrange bound
Alternating-tail bound

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FAQ

Questions

About the Calculus II: Sequences & Series sheet and making your own.

What does this Calculus II: Sequences & Series cheat sheet cover?: It has 11 sections: Sequences, Series Basics, p-Series, Divergence & Comparison, Integral Test, Ratio & Root, Alternating Series, Absolute Convergence, Power Series, Taylor & Remainder and Remainder. Together they cover the topics a typical Calculus II: Sequences & Series exam tests, including geometric series, p-series, divergence test and integral test.
Is the series convergence tests cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Calculus II: Sequences & Series cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Calculus II: Sequences & Series sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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