STMATH126 (Calculus 3) Exam 1 Flashcards (Definitions)

Flashcards for different convergence tests for infinite series problems.

Geometric Series

The general form is ∑ar^k. It converges when the absolute value of the common ratio |r| is less than 1 (0 < |r| < 1), with the sum being calculated using the formula a / (1 - r); it diverges if |r| ≥ 1. (Typically used for form ∑ar^k or ∑ar^k-1)

Telescoping Series

First write out the partial sums and then cancel the appropriate terms. Finally, take the limit as k approaches infinity of the resulting sum. (Typically used when partial fractions or terms cancel out [e.g. 1/k - 1/(k+1)])

Divergence Test

If the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. If the limit is zero, the test is inconclusive, and further analysis is required to determine convergence. (Typically used as a quick first step check)

Integral Test

The function must be positive, continuous, and decreasing for x ≥ 1 . If the improper integral converges, then the series converges; if the integral diverges, then the series diverges.

Comparison Test

You compare the series with a known benchmark series that is either convergent or divergent. If your series is less than a known convergent series, it converges; if it is greater than a known divergent series, it diverges. If neither condition is met, further analysis is needed. (Typically used when terms resembles p-series or known benchmark)

Limit Comparison Test

Take the series with an unknown convergence (a_k) and a series with a known convergence (b_k) that both have positive terms. Then calculate the limit as k approaches infinity of (a_k)/(b_k) = L. If L is positive and finite (L > 0), then both series either converge or diverge together; if L is zero and (b_k) converges, then (a_k) converges; if L is infinite and (b_k) diverges, then (a_k) diverges.

How do you apply the Alternating Series Test and interpret the results?

You must check two conditions for the series: first, the absolute value (cancel out (-1)^k+1) of the terms must be decreasing (0<a_k+1≤a_k), and second, the limit of (a_k) as k approaches infinity must be zero. If both conditions are satisfied, the series converges. (Typically used when the terms alternate in sign and decrease in magnitude. e.g. sin(k) or (-1)^k)

Conditional/Absolute Convergence

First check if the series converges absolutely by examining the convergence of the series of absolute values of the terms. If the absolute series converges, then the original series converges absolutely (Absolute convergence = Convergence). If the absolute series diverges but the original series converges, then it converges conditionally. If the series diverges, the absolute series also diverges. (Typically used when the terms alternate in sign and decrease in magnitude. e.g. sin(k) or (-1)^k)

Ratio Test

Calculate the limit of the absolute value of (a_n+1)/(a_n) as n approaches infinity. If this limit r exists, then: if r < 1, the series converges absolutely; if r > 1 or r is infinite, the series diverges; if r = 1, the test is inconclusive and further investigation is needed. (Typically used for series with factorials, exponentials, or products in the terms.)

Root Test

Calculate the limit of a_k to the k-th root as k approaches infinity. If this limit ρ exists, then: if 0 ≤ ρ < 1, the series converges absolutely; if ρ > 1 (or infinite), the series diverges; if ρ = 1, the test is inconclusive and requires further analysis. (Typically used when k appears in the terms of the series, especially with powers or exponentials.)

P-series Test

It shows that an infinite series 1/(n^p) converges for p > 1 and diverges for p ≤ 1. This test is useful for determining the convergence of series whose terms are in the form of 1/(n^p) where p is a constant.