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Is it a sequence or a series?
Sequence
Take the limit as n ( (
Series (Test for divergence)
Take the limit as n ( (
Limit exists
The sequence converges
Limit does not exist
The sequence diverges
Limit `" 0
The series diverges
Limit = 0
The series might converge. Run furthe≠ 0
The series diverges
Limit = 0
The series might converge. Run further tests.
What kind of series is it?
P-series
∑ 1/np
Telescoping series
∑ 1/n – 1/(n+1)
∑ 1/(n(n+1))
Alternating Series
∑ (−1)n bn
Geometric Series
∑ arn
p ≤ 1
The series diverges
p > 1
The series converges
|r| ≥ 1
The series diverges
|r| < 1
The series converges
Lim < 1
The series converges
Alternating Series Test
1) bn ≥ bn+1 for all n
2) limit of bn = 0
Fails test
?
Passes test
The series converges conditionally
If it is of the form
∑ 1/(n(n+1))
Use partial fractions to separate the terms in the denominator if necessary.
Find the pattern
Write out some terms until you see the pattern of recurrence. Then find the nth partial sum Sn
and take the limit as n ( (
Absolute value diverges
Use Alternating series test
Absolute value converges
The series converges
Check for Absolute Convergence
See if ∑ bn converges
Ratio Test
Useful for factorials (also power series)
Integral Test
The series converges if its integral converges (look for things that can be integrated, especially log’s)
Direct Comparison
Use when the comparison is in the right direction and easy
Limit Comparison
Works for almost all powers of n.
Comparison tests
Useful for powers of n
Lim > 1
The series diverges
Lim = 1
?
Lim is finite and non-zero
Both series converge or both series diverge
Lim is 0 or infinite/DNE
?
If series ( a convergent series, it converges
If series ≥ a divergent series, it diverges
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