Series Tests

Question 1

What does the Geometric Series Test state?

Answer 1

For Series of the form sigma n= 0 to infinity a((r)^n) = sigma n= 1 to infinity a((r)^n-1) where a is constant,

The series converges if |r| < 1, it converges to a/(1-r)

Question 2

What does the Divergence Test state?

Answer 2

For a series sigma a sub n, the series diverges if the limit as n approaches infinity of a sub n does not equal 0.

If the limit as n approaches infinity of a sub n equals 0, the test is inconclusive

This test indicates divergence, not convergence

Question 3

What does the Integral Test state?

Answer 3

1. If a sub n equals f sub n of x is continuous, positive, and decreasing, then:
2. the series, sigma n = b to infinity a sub n converges when the integral from b to infinity of f sub n dx converges.
3. The series diverges if the integral diverges

Note: If the integral converges to some value L, the series will not converge to the same value.

Question 4

What does the Direct Comparison Test state?

Answer 4

1. Let sigma n =k to infinity a sub n and sigma n = k to infinity b sub n be two series, and let a sub n be greater than or equal to 0 and less than or equal to b sub n.
2. If sigma b sub n converges, sigma a sub n also converges.
3. If sigma a sub n diverges, sigma b sub n diverges

Question 5

What does the Limit Comparison Test state?

Answer 5

1. a sub n and b sub n are greater than zero.
2. If the limit as n approaches infinity of a sub n over b sub n is greater than 0, then sigma a sub n and b sub n both converge or both diverge.
3. If the limit as n approaches infinity of a sub n over b sub n equals 0. and sigma b sub n converges, sigma a sub n converges.
4. If the limit as n approaches infinity of a sub n over b sub n equals infinity and sigma b sub n diverges. Sigma a sub n diverges

Question 6

What does the Absolute Convergence Test state?

Answer 6

If sigma absolute value of a sub n converges, then sigma a sub n converges absolutely

Question 7

What does the Root Test state?

Answer 7

1. Given sigma a sub n and the limit as n approaches infinity of the nth root of the abs value of a sub n raised to 1/n = S
2. If 0 < = S < =1, the series converges absolutely
3. If S = 1, the test is inconclusive
4. If S > 1, the series diverges

Question 8

What does the Ratio Test state?

Answer 8

Let sigma a sub n be the given infinite series, and let the limit as n approaches infinity of the abs value of (a sub n plus 1) / (a sub n) = R

2. If 0 <= R < 1, the series converges absolutely
3. If R = 1, the test is inconclusive
4. If R > 1, the series diverges

Question 9

What does the Alternating Series Test state?

Answer 9

The series sigma n =k to infinity (-1) raised to n times a sub n or sigma
n = k to infinity (-1) raised to n + 1 times a sub n converges exactly when:

1. a sub n is positive, decreasing, and the limit as n approaches infinity of a sub n = 0. If the third property specifically fails, then the series diverges by the Divergence Test