Series Flashcards
Describe the integral test
If an = f(n) where the function is a non-negative non-increasing function then the series an converges if and only if the integral of the function converges
Describe the limit comparison test
If the series an and bn are positive-termed series and the limit as n approaches infinity of an/bn = L where 0 < L < infinity, then either an and bn both converge or diverge
Describe the divergence test
If the limit as n approaches infinity of an doesn’t = 0, then the series of an diverges
Describe the ratio test
Let L = the limit as n approaches infinity of |an+1|/|an|
If L < 1, then an converges absolutely
If L > 1, or the limit goes to infinity, then an diverges
If L = 1 or if L does not exist, then the test fails
Decribe the alternating series test
When a series is alternating (-1)nbn, if bn > 0, bn+1 < bn, and the limit of bn as n approaches infinty is 0, then the series converges
Describe the root test
Let L = the limit as n approaches infinity of the nth root of |an|
If L < 1, then an converges absolutely
If L > 1, or the limit goes to infinity, then an diverges
If L = 1 or if L does not exist, then the test fails
In a geometric series, how do you find the value that series converges to?
If the series starts with n = 0, the the value of convergence is = a/(1 - r)
Describe the comparison test
Applies only to positive-term series
If an < bn and bn converges then an converges
If bn < an and bn diverges then an diverges
What is the limit as n approaches infinity for xn/n!?
0
What is the series representation for ex?
series as n approaches infinity of xn/n!
What is the series representation for e?
series as n approaches infinity of 1/n!
What is the limit as n approaches infinity of na/n?
1
