10.5

Question 1

What does the absolute convergence theorem state?

Answer 1

Is ∑|an| converges, ∑an converges

Note that not all convergent series need to be absolutely convergent, but if a series is absolutely convergent it must be convergent

Question 2

What does the ratio test say and when is it useful

Answer 2

the ratio test is useful for factorials and exponentials

You take the limit as n approaches infinity of absolute value of the second term in a series divided by the first term

If the result is less than 1 the series converges

If the result is more than 1 or is infinite, the series diverges

If the result = 1 the test is inconclusive

Question 3

What does the root test say and when is it useful

Answer 3

The root test is useful for nth powers

You take the limit as n approaches infinity of the nth root of the absolute value of a series

If the result is less than 1 the series converges

If the result is greater than 1 the series diverges

If the result = 1 the test is inconclusive