BC Calculus Unit 10 Infinite Series

nth term Test (Test for Divergence only)

If lim (n→∞) an ≠ 0, then the series diverges. If lim (n→∞) an = 0, then the nth term test is inconclusive.

Geometric Series Test

Series of the form ∑ a*r^n series converge if |r| < 1, diverges otherwise. Sum = a_1 / (1-r)

P-Series Test

For series of form ∑ 1/n^p the series converge if p > 1. Otherwise, series diverge. (p ≤ 1)

Integral Test

If an = f(x) is a Positive, Decreasing, Continuous Function, then ∑ an and ∫ f(x)dx either both converge or diverge. (good for functions that look like u-substitution problems)

Limit Comparison Test (LCT)

For an, bn > 0, (where bn is a comparison partner for an), if lim (n→∞) |an / bn| equals a finite positive number, then both series either converge or diverge.

Direct Comparison Test (DCT)

For an, bn > 0, (where bn is a comparison partner for an), if an ≤ bn and if ∑ bn converges, then ∑ an converges as well. If an ≥ bn and if ∑ bn diverges, then ∑ an diverges as well.

Alternating Series Test (AST)

If the series has the form ∑(-1)^n * an and meets two criteria: 1) lim (n→∞) |an| = 0 and 2) a(n+1) ≤ an then the series converge by AST.

Ratio Test

If lim (n→∞) |a(n+1) / an| < 1 then series converge. If lim (n→∞) |a(n+1) / an| > 1, series diverge. If equal to 1, then test is inconclusive. (Look for factorials or exponentials when determining if Ratio Test is a good fit)

Root Test

If lim (n→∞) (n√|an|) < 1 then series converge. If lim (n→∞) (n√|an|) > 1, series diverge. If equal to 1, then test is inconclusive. (Look to see if the entire expression is raised to an nth power to determine if good fit)

Telescoping Series

A series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and/or some of the last terms.

Alternating Series Remainder Theorem

For a convergent alternating series, |S − Sn| = |Rn| ≤ |a_(n+1)|

Maclaurin Series definition

∑ (f^(n)(0) / n!) * x^n from n=0 to ∞ = f(0) + f'(0)x + (f''(0) / 2!) * x^2 + (f'''(0) / 3!) * x^3 + …

Taylor Series definition

∑ (f^(n)(c) / n!) * (x − c)^n from n=0 to ∞ = f(c) + f'(c)(x − c) + (f''(c) / 2!) * (x − c)^2 + (f'''(c) / 3!) * (x − c)^3 + …

Maclaurin Series for sin x

x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + … + ((-1)^n * x^(2n+1)) / (2n + 1)!

Maclaurin Series for cos x

1 - (x^2 / 2!) + (x^4 / 4!) - (x^6 / 6!) + … + ((-1)^n * x^(2n)) / (2n)!

Maclaurin Series for e^x

1 + x + (x^2 / 2!) + (x^3 / 3!) + … + x^n / n!

Maclaurin Series for tan^-1(x)

x - (x^3 / 3) + (x^5 / 5) - (x^7 / 7) + … + ((-1)^n * x^(2n+1)) / (2n + 1)

Maclaurin Series for 1 / (1-x)

1 + x + x^2 + x^3 + … + x^n