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

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

3. 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

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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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→∞

12. 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

13. 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

14. 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! ...

15. 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! + ...

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

17. 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.