Convergence and Series Matt Z., Griffin, Jack

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

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)

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.

P-Series Test

Direct Comparison Test (DCT)

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

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

Root Test

Ratio Test

Series

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

The Four Known Series

Convergence of a Power Series If x is a variable, then an infinite series of the form ∑∞n=0 an= a0 + a1x + a2x2+...+ 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)

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: 0+1+2x+3x2+4x3+5x4 Find the General Term for f’(x): nxn-1

Lagrange Error Bound