Math 253 Series Convergence Cheat Sheet

The Math 253 Series Convergence Cheat Sheet is a document that provides a quick reference guide for students studying calculus. It includes information about different types of series and their convergence properties, helping students understand and solve problems related to series convergence.

FAQ

Q: What is series convergence?
A: Series convergence refers to the behavior of the sum of terms in a series.

Q: What is a convergent series?
A: A convergent series is a series whose terms add up to a finite value.

Q: What is a divergent series?
A: A divergent series is a series whose terms do not add up to a finite value.

Q: What is the comparison test for series convergence?
A: The comparison test states that if the terms of a series are always less than or equal to the terms of a convergent series, then the series is also convergent.

Q: What is the limit comparison test for series convergence?
A: The limit comparison test states that if the limit of the ratio of the terms of two series is a positive finite number, then the two series either both converge or both diverge.

Q: What is the ratio test for series convergence?
A: The ratio test states that if the limit of the absolute value of the ratio of consecutive terms in a series is less than 1, then the series is convergent.

Q: What is the root test for series convergence?
A: The root test states that if the limit of the nth root of the absolute value of the terms in a series is less than 1, then the series is convergent.

Q: What is the alternating series test for series convergence?
A: The alternating series test states that if the terms of a series alternate in sign and decrease in magnitude, then the series is convergent.

Q: What is the integral test for series convergence?
A: The integral test states that if a function f(x) is positive, continuous, and decreasing on an interval [a,∞), then the series ∑f(n) is convergent if and only if the improper integral ∫f(x)dx from a to infinity is convergent.

Q: What is the harmonic series?
A: The harmonic series is a divergent series whose terms are the reciprocals of the positive integers.