Series Tests

Question 1

What does the Geometric Series Test state?

Answer 1

Geometric Series Form: sigma n = 0 to infinity a*(r)^n

Series Converges if the absolute value of n is less than one

Series Diverges if the abs value of r greater than or equal to one

The sum of a convergent geometric series is sigma n = 0 to infinity a*(r)^n equals 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 equal 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. If the series sigma a sub n and sigma b sub n with 0 is less than or equal to a sub n and b sub n is greater than or equal to a sub n then:
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 = L
2. If L is less than one, converges
3. If L is greater than one, diverges
4. If L is equal to one, inconclusive

Question 8

What does the Ratio Test state?

Answer 8

Given sigma a sub n and the limit as n approaches infinity of the abs value of a sub n +1 over a sub n equals L.

2. If L is less than one, converges
3. If L is greater than one, diverges
4. If L is equal to one, inconclusive

Question 9

What does the Alternating Series Test state?

Answer 9

Given sigma -1 raised to n + 1 times U sub n and

1. U sub n all positive
2. U sub n is greater than or equal to (U sub n) + 1 for some n
3. U sub n approaches 0

sigma -1 raised to n + 1 times U sub n converges