Calc BC Series Tests and Conditions

Sigma notation of Geometric series

sigma(n=0, inf, a * r^n)

Conditions of Convergence of Geometric series

abs(r) < 1

Conditions of Divergence of Geometric series

abs(r) >= 1

Sigma notation for nth term test for Divergence

sigma(n=1, inf, a_n)

Conditions of Convergence of nth term test for Divergence

None

Conditions of Divergence of nth term test for Divergence

if lim(n→inf, a_n) ≠ 0, then series diverges

Sigma notation of Telescoping Series (not tested on AP)

sigma(n=1, inf, bn - b{n+1})

Conditions of Convergence of Telescoping Series (not tested on AP)

If in expanded form, the terms begin to "subtract out"

Conditions of Divergence of Telescoping Series (not tested on AP)

Not used for divergence

Sigma notation for Integral Test

sigma(n=1, inf, an), where an = f(n) >= 0

Conditions of Convergence of Integral Test

integral(1 to inf, f(x) dx) converges

Conditions of Divergence of Integral Test

integral(1 to inf, f(x) dx) diverges

Sigma notation of p-Series

sigma(n=1, inf, 1 / n^p)

Conditions of Convergence of p-Series

p > 1

Conditions of Divergence of p-Series

p <= 1

Sigma notation for Direct Comparison Test

sigma(n=1, inf, a_n)

Conditions of Convergence of Direct Comparison

0 < an

Conditions of Divergence of Direct Comparison

0 < bn

Sigma notation for Limit Comparison

sigma(n=1, inf, a_n)

Conditions of Convergence of Limit Comparison

lim(n→inf, an / bn) = L > 0 and sigma(n=1, inf, b_n) converges

Conditions of Divergence of Limit Comparison

lim(n→inf, an / bn) = L > 0 and sigma(n=1, inf, b_n) diverges

Sigma notation for Alternating Series

sigma(n=1, inf, (-1)^(n+1) * a_n)

Conditions of Convergence of Alternating Series

0 < a(n+1)

Conditions of Divergence of Alternating Series

Not used for divergence

Sigma notation for Ratio Test

sigma(n=1, inf, a_n)

Conditions of Convergence of Ratio Test

lim(n→inf, abs(a(n+1) / an)) < 1

Conditions of Divergence of Ratio Test

lim(n→inf, abs(a(n+1) / an)) > 1

Sigma notation for Root Test (not tested on AP)

sigma(n=1, inf, a_n)

Conditions of Convergence of Root Test (not tested on AP)

lim(n→inf, nthroot(n, abs(an))) < 1

Conditions of Divergence of Root Test (not tested on AP)

lim(n→inf, nthroot(n, abs(an))) > 1