AP Calculus BC Series Tests Exam

Nth Term Test (NTT) - ANS 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) - ANS 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) - ANS 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) - ANS 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) - ANS 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) - ANS 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 - Use when other tests do not directly apply but a similar series has a known convergence/divergence Limit Comparison Test (LCT) - ANS Take the limit as n - inf. of a[n] divided by a comparison function with the same highest degree in the numerator and denominator. - If limit converges, series converges - If limit diverges, series diverges - Use when other tests do not apply and DCT fails Alternating Series Test (AST) - ANS ((-1)^n)(a[n]), take the limit as n - inf. of the attached a[n] and verify its decrease. - If limit = 0 and a[n+1] = a[n], series converges - If limit != 0, series diverges like NTT - If limit = 0 but a[n+1] != a[n], test is inconclusive If series converges, evaluate the series without (-1)^n. - If a[n] converges, series converges absolutely - If a[n] diverges, series converges conditionally - Use when series involves (-1)^n Ratio Test (RaT) - ANS Take the limit as n - inf. of abs(a[n+1]/a[n]). - If limit 1, series converges absolutely - If limit 1, series diverges - If limit = 1, test is inconclusive - Use when other tests do not apply and series appears to decrease Root Test (RoT) - ANS Take the limit as n - inf. of the nth root of abs(a[n]). - If limit 1, series converges absolutely - If limit 1, series diverges - If limit = 1, test is inconclusive - Use when other tests do not apply and series is to the nth power