Calc III Test 2

Question 1

What does the Ratio Test state?

Answer 1

series ∑aₙ, if ρ = lim |aₙ₊₁ / aₙ|

Question 2

if p<1

Answer 2

absolutely convergent

Question 3

p>1

Answer 3

divergent

Question 4

p=1

Answer 4

inconclusive

Question 5

What does it mean for a series to be absolutely convergent

Answer 5

A series ∑aₙ is absolutely convergent if ∑|aₙ| converges.

Question 6

What is the alternating series estimation theorem

Answer 6

For an alternating series that converges, the error in approximating with the first N terms is ≤ the (N+1)th term.

Question 7

When does an alternating series converge?

Answer 7

If the terms decrease in absolute value and approach 0.

Question 8

How do you find the interval of convergence for a power series?

Answer 8

Use the Ratio Test to find where ρ < 1, then test endpoints separately.

Question 9

What type of series is ∑(2x−3)ⁿ/n² and how to test its convergence?

Answer 9

It’s a power series; use the Ratio Test and check endpoints with known convergence tests like p-series or alternating series.

Question 10

What is the general form of a Taylor series for f(x) centered at a?

Answer 10

∑(f⁽ⁿ⁾(a)/n!) · (x − a)ⁿ

Question 11

What’s the Taylor series for cos(x)?

Answer 11

∑(−1)ⁿ x²ⁿ / (2n)!

Question 12

How do you find the Taylor series for cos(√x)?

Answer 12

Substitute u = √x into the series for cos(u).

Question 13

How to find an upper bound for error in alternating series?

Answer 13

Use the first omitted term: |Error| ≤ |aₙ₊₁|

Question 14

What’s the max error for approximating ln(1 + x) with x − x²/2 + x³/3 − x⁴/4 in (−½, ½)?

Answer 14

≤ x⁵ / 5, and for x in (−½, ½), that’s at most 1/160.

Question 15

What’s the 2nd degree Taylor polynomial for √x at a = 4?

Answer 15

Use f(x) = x¹ᐟ² and its first 2 derivatives at x = 4 to get:
P₂(x) = 2 + ¼(x − 4) − 1/64(x − 4)²