Calc III

Sequence

An ordered and infinite list of numbers: {a_n} = {a₁, a₂, a₃, a₄, … , a_n,.. }

Plan of Attack for converging vs diverging sequences

1. Use limit laws as a function of f(x).

   1. Like Limit Laws, L'Hôpital's Rule, and Intermediate Forms

2. Squeeze Theorem. ALso Check for alternation to use

Intermediate Rule

When you see 1^∞, 0^0, ∞^0 Then ln(L)=lim g(n) * ln(f(n)) Then after do L=e^lnL

If a sequence is bounded and monotonic, then the sequence

Coverges

Geonometric Series

Telescopic Series

Every term except first and last cancel out

P-Series

1/n^p

• Converges if p > 1

• Diverges if p ≤ 1

Alternating Series

Divergence Test

if limiit to inf ≠ 0, it diverges

Comparison Test

If both a and b are positive and a <= b, b convg = a convg | a div = b div

Theorem-limit comparison Test

Absolute Convergence Test

Just put a absolute value around it

Integral Test

Ratio Test

Use when we see factors of x^_n or a factorial !.

Root Test

Use when you see xⁿ

Plan of Attack

1. Geometric Check

2. Telescopic Check

3. P-Series Check

4. Divergence Test

5. Comparison Test

6. Limit Comp

7. Integral Test

8. Altenration

9. Abs conv

10. Ratio

11. Root

Practice Problems

Just do all of 11.7

Power Series

Must also always chekc endpoints

What is this series equal to

Basic geonometric series to equation

Taylor and MacLaurin series