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 Maclaurin Series

ex = ∑ xn/n!
= 1 + x + x²/2! + x³/3! + ...
sin(x) = ∑ (-1)nx2n+1/(2n+1)!
= x - x³/3! + x⁵/5! - ...
cos(x) = ∑ (-1)nx2n/(2n)!
= 1 - x²/2! + x⁴/4! - ...
(1+x)k = ∑ (k choose n)xn
= 1 + kx + k(k-1)x²/2! + ...

Quick Reference

Test Checklist:
1. Check if it's geometric (easy win!)
2. Try ratio test for factorials/exponentials
3. Use root test for nth powers
4. Check endpoints for power series
5. Remember: convergence ≠ absolute convergence