AP Calculus BC Series Tests
Nth Term Test (NTT) - Correct Answers -Take the limit as n -> inf. of a[n].
- If limit != 0, series diverges
- If limit = 0, test is inconclusive
- Use when a[n] appears to not approach 0
Telescoping Series Test (TST) - Correct Answers -Break a[n] down using PFD, plugging
in n to get terms of a[n]. Use parts that do not cancel out to create S[n], then take the
limit as n -> inf. of S[n].
- If limit converges, series converges with sum of limit
- Use when a[n] can be broken down using PFD
Geometric Series Test (GST) - Correct Answers -a[n] = a(r)^n, where a is the coefficient
and r is the ratio to the nth power.
- If abs(r) < 1, series converges with sum of a/(1-r)
- If abs(r) >= 1, series diverges
- Use when a[n] is a constant to the nth power
Integral Test (IT) - Correct Answers -If a[n] = f(x) is continuous, positive (f(x) = (+)), and
decreasing (f'(x) = (-)) for x >= 1, then take the integral from 1 to inf. of f(x).
- If integral converges, series converges
- If integral diverges, series diverges
- Use when other tests do not apply and IT rules can be met
P-series Test (PST) - Correct Answers -a[n] = k/n^p, where k is a constant and p is a
real number.
- If p > 1, series converges
- If 0 <= p < 1, series diverges
- If p = 1, series diverges harmonically
- Use when a[n] is a constant over n^p
Direct Comparison Test (DCT) - Correct Answers -Compare a[n] to a simpler, similar
series. the lesser of the two becomes a[n], while the greater of the two becomes b[n].
Determine the convergence/divergence of the simpler series.
- If b[n] converges, a[n] converges
- If a[n] diverges, b[n] diverges
Nth Term Test (NTT) - Correct Answers -Take the limit as n -> inf. of a[n].
- If limit != 0, series diverges
- If limit = 0, test is inconclusive
- Use when a[n] appears to not approach 0
Telescoping Series Test (TST) - Correct Answers -Break a[n] down using PFD, plugging
in n to get terms of a[n]. Use parts that do not cancel out to create S[n], then take the
limit as n -> inf. of S[n].
- If limit converges, series converges with sum of limit
- Use when a[n] can be broken down using PFD
Geometric Series Test (GST) - Correct Answers -a[n] = a(r)^n, where a is the coefficient
and r is the ratio to the nth power.
- If abs(r) < 1, series converges with sum of a/(1-r)
- If abs(r) >= 1, series diverges
- Use when a[n] is a constant to the nth power
Integral Test (IT) - Correct Answers -If a[n] = f(x) is continuous, positive (f(x) = (+)), and
decreasing (f'(x) = (-)) for x >= 1, then take the integral from 1 to inf. of f(x).
- If integral converges, series converges
- If integral diverges, series diverges
- Use when other tests do not apply and IT rules can be met
P-series Test (PST) - Correct Answers -a[n] = k/n^p, where k is a constant and p is a
real number.
- If p > 1, series converges
- If 0 <= p < 1, series diverges
- If p = 1, series diverges harmonically
- Use when a[n] is a constant over n^p
Direct Comparison Test (DCT) - Correct Answers -Compare a[n] to a simpler, similar
series. the lesser of the two becomes a[n], while the greater of the two becomes b[n].
Determine the convergence/divergence of the simpler series.
- If b[n] converges, a[n] converges
- If a[n] diverges, b[n] diverges
