Guide to Infinite Series Tests |
Table of Tests to use on Infinite Series: (see notes below)
Test | Series | Convergence or Divergence | Comments |
nth Term | Diverges if | Inconclusive if lim = 0 | |
Geometric Series |
(i) Converges if | r | < 1 (ii) Diverges if | r | 1 |
If convergent, the sum is
Useful for comparison tests | |
p-series | (i) Converges if p > 1 (ii) Diverges if p 1 |
Useful for comparison tests | |
Integral | a_{n} = f(x) |
(i) Converges if converges (ii) Diverges if diverges |
f (x) must be continuous, positive, decreasing. |
Comparison Test Limit Comparison Test |
test , select both positive |
(i) converges if converges and for all values of n. (ii) diverges if diverges and for all values of n. (iii) If , then both series converge or both series diverge. |
The comparison series is often a geometric or a p-series. |
Ratio Test | If ),: (i) converges if L < 1 (ii) diverges if L > 1 |
Inconclusive if L = 1. Useful if a_{n} involves factorials or nth powers If all a are > 0, ignore the absolute value signs. | |
Root Test | If , the series: (i) converges if L < 1 (ii) diverges if L > 1 |
Inconclusive if L = 1. Useful if a_{n} includes only nth powers If all a's are > 0, ignore the absolute value sign. | |
Alternating Series Test | a_{n} > 0 |
Converges if a_{k} a_{ k+1} for every k and |
Applies only to an alternating series. |
Absolute Convergence Test | If is convergent then is convergent |
Useful for series that contain both positive & negative terms. |
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Notes:
Because there are so many different tests, this topic can be a major brain-strain unless we develop some strategies to approach the questions! Here then is a list of hints that might make things a little easier.
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