Tests for Sequences and Series
Multiple Choice
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40 Terms
Nth Term
an
What does the nth term test for?
divergence
Nth Term diverges conditions
lim a ≠ 0
Geometric Series
arn
Geometric Series Convergence Conditions
l r l < 1
Geometric Series Divergence Conditions
l r l ≥ 1
What does the Geometric Series converge to?
the sum; S = a / (1-r)
Telescoping Series
(an - an+1)
Telescoping Series Convergence Conditions
lim an = L; L has to be a finite number
What does the telescoping Series converge to?
the sum; S = a1 - L
P-series test
1 / np
P-series test convergence conditions
p > 1
P-series test divergence conditions
0 < p ≤ 1
What if p = 1 in a p-series test?
divergent harmonic series
Alternating Series Test
(-1)n * an or (-1)n+1 * an+1
Alternating Series Test convergent conditions
0 < an+1 ≤ an (nonincreasing) & lim an = 0
Alternating Series Test Remainder
l S - Sn l = l Rn l ≤ an+1
Integral Test conditions for f(x)
positive, continuous, and decreasing
Integral Test
an = f(n)
Integral Test Convergence Conditions
∫ f(x) dx converges, then series converges
Integral Test Divergence Conditions
∫ f(x) dx diverges, then series diverges
Ratio Test
an
Ratio Test Convergence Conditions
lim l an+1 / an l < 1
Ratio Test Divergence Conditions
lim l an+1 / an l > 1 or = ∞
Ratio Test is inconclusive if?
lim l an+1 / an l = 1
Root Test
an
Root Test Convergence Conditions
lim n√l an l < 1
Root Test Divergence Conditions
lim n√l an l > 1
The root test is inconclusive when?
lim n√l an l = 1
Direct Comparison Test
an
Direct Comparison Test Convergence Conditions
0 < an ≤ bn and bn converges
Direct Comparison Test Divergence Conditions
0 < bn ≤ an and bn diverges
Limit Comparison Test
an
Limit Comparison Test Convergence Conditions
lim (an / bn) = L > 0 & bn converges
Limit Comparison Test Divergence Conditions
lim (an / bn) = L > 0 & bn diverges
L must be what in limit comparison test?
finite and positive
Absolute vs. Conditional Convergence
once identified as convergence, use tests to see if l an l is convergent; yes- absolute or no- conditional
Sequence
a list; ex: 2, 4, 6, 8
Series
ex: 2 + 4 + 6 + 8
Recursive Definition (Implicit Definition) vs. Explicit Definition for Sequences
recursive- an+1 = an + x; explicit- an = 3n - 2
