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MAT 266 Chapter 8: Sequences & Series Cheat Sheet

MAT 266 Chapter 8: Sequences & Series Cheat Sheet





Sequences
Series
Convergence
Power Series
Taylor/Maclaurin





MAT 266 Chapter 8 Cheat Sheet

Sequences • Series • Convergence • Power Series • Taylor/Maclaurin





8.1 Sequences



Sequence Behavior


For sequence an:
• Converges if lim(n→∞) an = L (finite)
• Diverges if limit doesn't exist or is ±∞
• Oscillates if it alternates between values




Common Limits:
• lim(n→∞) 1/n = 0
• lim(n→∞) rn = 0 if |r| < 1
• lim(n→∞) rn = ∞ if r > 1








8.2 Infinite Series



Series Convergence


∞
∑ an = a1 + a2 + a3 + ...
n=1

Converges if lim(n→∞) Sn = L (finite)
where Sn = partial sum = a1 + a2 + ... + an




Geometric Series:
∞
∑ arn-1 = a + ar + ar² + ar³ + ...
n=1

Converges to a/(1-r) if |r| < 1
Diverges if |r| ≥ 1








8.4 Convergence Tests




Ratio Test


L = lim(n→∞) |an+1/an|



• L < 1: Converges absolutely
• L > 1: Diverges
• L = 1: Test fails






Root Test


L = lim(n→∞) ⁿ√|an|



• L < 1: Converges absolutely
• L > 1: Diverges
• L = 1: Test fails






Integral Test


If f(x) ≥ 0, decreasing, continuous



∞ ∞
∑ f(n) and ∫ f(x)dx
n=1 1
both converge or both diverge






Comparison Test


0 ≤ an ≤ bn for all n



• If ∑bn converges → ∑an converges
• If ∑an diverges → ∑bn diverges










8.5 Power Series



Interval & Radius of Convergence


∞
∑ cn(x - a)n = c0 + c1(x-a) + c2(x-a)² + ...
n=0




Radius of Convergence R:
Using Ratio Test: R = lim(n→∞) |cn/cn+1|
Or: R = 1/lim(n→∞) |cn+1/cn|

Interval: (a-R, a+R)
Check endpoints separately!








8.6 Functions as Power Series




Basic Series


1/(1-x) = ∑ xn for |x| < 1

1/(1+x) = ∑ (-1)nxn for |x| < 1






Substitution Method


Replace x with u in basic series

Example: 1/(1-x²)
= ∑ (x²)n = ∑ x2n






Integration Method


ln(1+x) = ∫ 1/(1+x) dx
= ∫ ∑(-1)nxn dx
= ∑ (-1)nxn+1/(n+1)






Differentiation Method


If f(x) = ∑ cnxn
then f'(x) = ∑ ncnxn-1










8.7 Taylor & Maclaurin Series



General Formulas


Taylor Series (about x = a):
f(x) = ∑ f(n)(a)/n! · (x-a)n
= f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...




Maclaurin Series (about x = 0):
f(x) = ∑ f(n)(0)/n! · xn
= f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...






Common M…