1. Convergence and Series
Matt Z., Griffin, Jack

2. Series
Telescopic Series: Write it out as partial fractions, will converge if the terms collapse (cancel out)
Ex.
Properties of Convergent Series: ∑an= A ∑bn= B
∞∑n=1 C * an = C* ∞∑n=1 an = C*A
∞∑n=1 (an +/- bn) = ∞∑n=1an +/- ∞∑n=1 bn = A +/- B

3. Nth Term Test Only tests divergence
If limn→ ∞an ≠ 0, then ∞∑n=1 an diverges
If limn→ ∞an = 0, then inconclusive (must use a different test)

4. Integral Test
If an is a positive sequence and an = f(n) where f(n) is a continuous, positive, decreasing function, then:
∞∑n=N an and ∞∫N f(x)dx either both diverge or both converge
If they converge, it may not be the same value for both functions.

5. P-Series Test

6. Direct Comparison Test (DCT)

7. Limit Comparison Test
If an > 0 and bn > 0 for all n ≥ N (where N is a positive integer),
If limn→ ∞ (an/bn )= C, where 0 < C < ∞, then A and B both converge or both diverge
If limn→ ∞ (an/bn )= 0, then if ∞∑n=1 bn converges then ∞∑n=1 an converges
If limn→ ∞ (an/bn )= ∞, then if ∞∑n=1 bn diverges then ∞∑n=1 an diverges

8. Alternating Series Test (AST)
∞∑n=1 (-1)n an
If limn→ ∞ an= 0, and an+ 1 ≤ an for all n, then it converges
If it doesn’t pass these tests, then it diverges

9. Root Test

10. Ratio Test

11. Series

12. Taylor & MacLaurin Series
They are used to approximate the behavior of a function
MacLaurin polynomials are a special case of a Taylor polynomial centered at x=0

13. The Four Known Series

14. Convergence of a Power Series
If x is a variable, then an infinite series of the form ∑∞n=0 an= a0 + a1x + a2x anxn is called a power series
Interval of Convergence: a set of x-values for which the power series converges
Radius of Convergence: the distance from the mid-point to the end point of the IOC
3 cases: 1) Converges at the center ROC=0 2) Converges for all x ROC=∞
3) Converges on an interval ROC=.5(IOC)

15. Derivatives and Integrals of Power Series
f(x) = 1/(1-x)
1st five terms: 1+x+x2+x3+x4
General Term: xn
Integrate the terms: c+x+.5x2+.333x3+.25x4+.2x5
Find the General Term for ഽf(x): (1/n+1)(xn+1)
f(x) = 1/(1-x)
1st five terms: 1+x+x2+x3+x4
General Term: xn
Derive the terms: x+3x2+4x3+5x4
Find the General Term for f’(x): nxn-1

16. Lagrange Error Bound