Connor Curran, Cole Davidson, Sophia Drager, Avash Poudel Series and Error Connor Curran, Cole Davidson, Sophia Drager, Avash Poudel

Series Tests Nth term Geometric Direct Comparison P-Series Integral Limit Comparison Ratio Root Geometric P-Series Harmonic Telescoping Alternating Nth term

Geometric Used when: a constant is raised to the power of n Converge if: |r| < 1 Diverge if: |r| ≥ 1 If convergent, series will converge to a1 / 1 - r

P-Series 1 / np Convergent if: p > 1 Divergent if: p < 1 Cannot determine what the series converges to

Harmonic Series 1 / n Diverges because it is a p-series with p = 1 Alternating harmonic series converges because of Alternating Series Test

Telescopic Series A series whose partial fraction form collapses to a finite series If terms collapse, series is convergent Series converges to the finite series after the collapse

Alternating Series with alternating signs Converge if: Terms are strictly alternating in sign every other term |bn+1| < |bn| (decreasing in magnitude) Limit is equal to 0 Sum can be estimated using using a partial sum

Nth Term Test Divergent if: limit is not equal to 0 If limit is equal to 0, test is inconclusive and another test is required Value of limit is not sum of the series

Direct Comparison Test Use when a series closely resembles a simpler series Let 0<an≤bn If bn converges, then an converges If an diverges, then bn diverges ∞ n=1 Σ ∞ n=1 Σ ∞ n=1 Σ ∞ n=1 Σ

Integral Test Series must be positive, continuous, and decreasing Convergent if: integral from from 1 to infinity is convergent Divergent if: integral from 1 to infinity is divergent Does not determine what series converges to

Limit Comparison Test If an > 0 and bn > 0 for all n If lim an/bn = c, 0<c<∞, then an and bn both converge or diverge If lim an /bn = 0, then if bn converges an converges If lim an/bn = ∞, then if bn diverges an diverges n→∞ ∞ n=1 Σ ∞ n=1 Σ ∞ n=1 Σ ∞ n=1 Σ n→∞ ∞ n=1 Σ ∞ n=1 Σ n→∞

Ratio Test Convergent if: limit of |an+1 / an| < 1 Divergent if: limit of |an+1 / an| > 1 Inconclusive if: limit of |an+1 / an| = 1

Root Test Convergent if: limit of nth root of |an| < 1 Divergent if: limit is > 1 or equal to infinity Inconclusive if: limit is equal to 1

Taylor Series Used to approximate the behavior of a function; generated by f at x=a P(x) = f(a) + f’(a)*(x-a)’ + f’’(a)*(x-a)2/2! + f’’’(a)*(x-a)3/3! ...

Maclaurin Series A special case of Taylor series generated by f at x=0 P(x) = f(0) + f’(0)*x1 + f’’(0)*x2/2! + f’’’(0)*x3/3! + ...

Alternating Series Error If a series satisfies the alternating series test |Error| ≤ an+1, where Sn is the nth partial sum of the series.

Lagrange Error Bound When a Taylor series is used, the error can be approximated by using the Lagrange Error Bound |Rn(x)| ≤ |fn+1(z)(x-c)n+1/(n+1)!| where z is the number that maximizes the n+1 derivative between x and c, x is the estimate, n is the degree of the taylor polynomial, and c is the center.