Series and Sequences Cheat Sheet

A printable reference covering sequence notation, arithmetic and geometric series, convergence tests, power series, and Taylor series for grades 11-12.

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Calculus: Series and Sequences covers how ordered lists of numbers and infinite sums behave. Students need this cheat sheet because series questions often require choosing the right test before doing any algebra. It helps connect patterns, limits, convergence, and function approximations in one printable reference. These ideas are essential for AP Calculus BC and first-year college calculus.

Key Facts

A sequence $\{a_n\}$ converges to $L$ if $\lim_{n \to \infty} a_n = L$ .
An infinite series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ has a finite limit.
The geometric series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1-r}$ when $|r| < 1$ and diverges when $|r| \ge 1$ .
The $p$ -series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges when $p > 1$ and diverges when $p \le 1$ .
The divergence test says that if $\lim_{n \to \infty} a_n \ne 0$ or the limit does not exist, then $\sum a_n$ diverges.
The ratio test uses $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$ ; the series converges if $L < 1$ , diverges if $L > 1$ , and is inconclusive if $L = 1$ .
A power series $\sum_{n=0}^{\infty} c_n(x-a)^n$ converges for $|x-a| < R$ , diverges for $|x-a| > R$ , and must be checked separately at $x=a \pm R$ .
The Taylor series for a function centered at $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ .

Vocabulary

Sequence: A sequence is an ordered list of numbers written as $\{a_n\}$ , where $n$ usually represents a positive integer position.
Series: A series is the sum of the terms of a sequence, written as $\sum_{n=1}^{\infty} a_n$ for an infinite series.
Partial Sum: A partial sum is the finite sum $S_N = \sum_{n=1}^{N} a_n$ used to study whether an infinite series converges.
Convergence: Convergence means a sequence or series approaches a finite value as $n \to \infty$ .
Radius of Convergence: The radius of convergence $R$ is the distance from the center $a$ within which a power series $\sum c_n(x-a)^n$ converges.
Taylor Series: A Taylor series represents a function near $x=a$ using derivatives in the form $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ .

Common Mistakes to Avoid

Using the divergence test to prove convergence is wrong because the test only proves divergence when $\lim_{n \to \infty} a_n \ne 0$ or does not exist.
Forgetting endpoint checks for a power series is wrong because the ratio or root test usually gives only $|x-a| < R$ , while $x=a-R$ and $x=a+R$ can behave differently.
Treating every alternating series as convergent is wrong because the alternating series test also requires $b_n \to 0$ and the positive terms $b_n$ to eventually decrease.
Applying the geometric sum formula when $|r| \ge 1$ is wrong because $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ only holds for $|r| < 1$ .
Ignoring absolute convergence is wrong because a series may converge conditionally, and tests like the ratio test often determine whether $\sum |a_n|$ converges.

Practice Questions

1 Determine whether $\sum_{n=1}^{\infty} \frac{3}{n^2}$ converges or diverges, and name the test or rule used.
2 Find the sum of the geometric series $\sum_{n=0}^{\infty} 5\left(\frac{2}{3}\right)^n$ .
3 Find the radius and interval of convergence for $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$ .
4 Explain why checking $\lim_{n \to \infty} a_n = 0$ is not enough to conclude that $\sum_{n=1}^{\infty} a_n$ converges.

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