Formulas and Tests for Sequences and Series

Arc length of a function y = f(x)

L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx

Arc length of a function x = g(y)

L = \int_c^d \sqrt{1 + (g'(y))^2} \, dy

Surface area of revolution around the x-axis for y = f(x)

S = 2\pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} \, dx

Fluid force on a submerged vertical surface

Fluid force F = \int_a^b \rho g h(y) w(y) \, dy, where \rho is the fluid density, g is gravitational acceleration, h(y) is the depth, and w(y) is the width at y.

Sequence

A sequence is an ordered list of numbers, often defined by a formula for the n-th term, denoted as a_n.

Convergence of a sequence

A sequence a_n converges to a limit L if, for every \epsilon > 0, there exists an N such that |a_n - L| < \epsilon for all n > N.

Series

A series is the sum of the terms of a sequence, often written as \sum_{n=1}^{\infty} a_n.

Difference between a sequence and a series

A sequence is an ordered list of terms, while a series is the sum of a sequence of terms.

Partial sum of a series

The partial sum S_n is the sum of the first n terms of a series: S_n = \sum_{k=1}^n a_k.

Geometric series

A geometric series is a series where each term is a constant multiple of the previous term, given by \sum_{n=0}^{\infty} ar^n where a is the first term and r is the common ratio.

Convergence criteria for a geometric series

A geometric series \sum_{n=0}^{\infty} ar^n converges if |r| < 1 and diverges if |r| \geq 1.

n-th term test for divergence

If \lim_{n \to \infty} a_n \neq 0 or does not exist, then the series \sum a_n diverges.

p-series test

A p-series \sum \frac{1}{n^p} converges if p > 1 and diverges if p \leq 1.

Comparison test for series with positive terms

If 0 \leq a_n \leq b_n for all n and \sum b_n converges, then \sum a_n converges. If \sum a_n diverges, then \sum b_n diverges.

Limit comparison test

For series \sum a_n and \sum b_n with positive terms, if \lim_{n \to \infty} \frac{a_n}{b_n} = c where 0 < c < \infty, then \sum a_n and \sum b_n either both converge or both diverge.

Ratio test

For a series \sum a_n, if \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L: - L < 1: the series converges absolutely. - L > 1: the series diverges. - L = 1: the test is inconclusive.

Absolute convergence

A series \sum a_n converges absolutely if \sum |a_n| converges.

Conditional convergence

A series \sum a_n converges conditionally if \sum a_n converges but \sum |a_n| diverges.

Alternating series test (Leibniz's test)

For an alternating series \sum (-1)^{n-1} b_n, if b_n is decreasing and \lim_{n \to \infty} b_n = 0, then the series converges.

Integral test

If f(x) is positive, continuous, and decreasing for x \geq 1, then the series \sum_{n=1}^{\infty} f(n) and the integral \int_1^{\infty} f(x) \, dx either both converge or both diverge.