Infinite Sequences and Series: Convergence Tests, Taylor and Maclaurin Series

An infinite sequence is an ordered list of numbers, denoted as {a_n} = a_1, a_2, a_3, ..., a_n, .... A sequence converges if the limit of its terms as n approaches infinity exists and is a finite number L (lim (n->inf) a_n = L). Otherwise, the sequence diverges.

An infinite series is the sum of the terms of an infinite sequence, denoted as sum (n=1 to inf) a_n = a_1 + a_2 + a_3 + .... A series converges if the sequence of its partial sums S_N = sum (n=1 to N) a_n converges to a finite limit S (lim (N->inf) S_N = S). Otherwise, the series diverges.

If lim (n->inf) a_n does not equal 0, then the series sum (n=1 to inf) a_n diverges. If lim (n->inf) a_n = 0, the test is inconclusive; the series may converge or diverge.

If f is a positive, continuous, and decreasing function for x >= 1, and a_n = f(n), then the series sum (n=1 to inf) a_n and the improper integral integral (1 to inf) f(x) dx either both converge or both diverge.

A p-series is a series of the form sum (n=1 to inf) 1/n^p. It converges if p > 1 and diverges if p <= 1.

Suppose that sum a_n and sum b_n are series with positive terms. If 0 <= a_n <= b_n for all n greater than some N: 1. If sum b_n converges, then sum a_n converges. 2. If sum a_n diverges, then sum b_n diverges.

Suppose that sum a_n and sum b_n are series with positive terms. If lim (n->inf) (a_n / b_n) = L, where L is a finite, positive number (0 < L < inf), then either both series converge or both series diverge.

An alternating series is a series whose terms alternate in sign, typically of the form sum (n=1 to inf) (-1)^(n-1) b_n or sum (n=1 to inf) (-1)^n b_n, where b_n > 0. The series converges if both conditions are met: 1. lim (n->inf) b_n = 0. 2. b_n is a decreasing sequence (i.e., b_(n+1) <= b_n for all n greater than some N).

A series sum a_n is absolutely convergent if the series of absolute values sum |a_n| converges. If sum a_n converges but sum |a_n| diverges, then the series sum a_n is conditionally convergent.

For a series sum a_n, let L = lim (n->inf) |a_(n+1) / a_n|. 1. If L < 1, the series converges absolutely. 2. If L > 1 or L = inf, the series diverges. 3. If L = 1, the test is inconclusive.

For a series sum a_n, let L = lim (n->inf) |a_n|^(1/n). 1. If L < 1, the series converges absolutely. 2. If L > 1 or L = inf, the series diverges. 3. If L = 1, the test is inconclusive.

A power series is a series of the form sum (n=0 to inf) c_n (x-a)^n, where x is a variable and a is the center of the series. For a given power series, there are three possibilities for convergence: 1. The series converges only at x = a. 2. The series converges for all x (R = inf). 3. There is a positive number R (radius of convergence) such that the series converges if |x-a| < R and diverges if |x-a| > R. The interval of convergence must be checked at the endpoints x = a +/- R.

If a function f has derivatives of all orders at x = a, then the Taylor series for f(x) centered at a is given by the sum of its derivatives evaluated at a, divided by n! and multiplied by (x-a)^n.

A Maclaurin series is a special case of a Taylor series where the series is centered at a = 0.