8.1-8.6 sequences/series tests

Question 1

What is the Absolute Value Theorem?

Answer 1

If the lim_n→∞|a_n|=0, then the overall limit is 0

Question 2

What is a Telescoping Series? When will it converge? What is the sum?

Answer 2

Where most of its values cancel. It will converge if and only if a_n approaches a finite number, where the sum is a_n-lim_n→∞a_n+1

Question 3

What is a Geometric Series? When will it converge? What is the sum?

Answer 3

Where terms are of the form a(r)ⁿ, a being the initial term. It will converge when |r| is between (0,1), where the sum is a/(1-r)

Question 4

What is the nth Term Test?

Answer 4

If the lim_n→∞a_n≠0, then the series diverges.
However, just because the nth Term approaches 0 does not mean that it converges.

Question 5

What is the Integral Test? When does it apply?

Answer 5

Σ_n=1^∞a_n and ∫₁^∞f(x)dx for f(x)=a_x either both converge or both diverge. Must state that it applies when f(x) is positive, continuous, and decreasing for x≥1 (or whatever n starts from). Note that the bounds of the integral are important!

Question 6

What is a P-series? When will it converge?

Answer 6

Where the terms are of the form 1/n^p. It converges if p>1 and diverges if 0<p≤1.

Question 7

What is the Maximum Error?

Answer 7

If the series converges, the remainder is inclusively bounded by 0 and ∫_n^∞f(x)dx

Question 8

What is the Direct Comparison Test?

Answer 8

0<smaller series≤largers series
If the larger series converges, the smaller series must also converge.
If the smaller series diverges, the larger series must also diverge.
Must state the relationship between the two series and for what numbers n, as well as why the comparative series converges/diverges.

Question 9

What is the Limit Comparison Test? When does it apply?

Answer 9

If there exists the solely positive sequences a_n and b_n lim_n→∞a_n/b_n is equal to a finite and positive number, then the sequences will either both converge or diverge

Question 10

What is the Alternating Series Test?

Answer 10

The series Σ_n=1^∞(-1)ⁿa_n converges if (1) the series is non-increasing and (2) lim_n→∞a_n=0 (whether just the a_n converges determines conditional or absolute convergence)

Question 11

What is the Alternating Series Remainder?

Answer 11

If the series is non-increasing, |R| is less than or equal to the first term outside of the partial sum.

Question 12

What is the Ratio Test?

Answer 12

If lim_n→∞|a_n+1/a_n|
is <1, the series converges (absolutely)
is >1, the series diverges
is =1, inconclusive

Question 13

What is the Root Test?

Answer 13

If lim_n→∞|a_n|^1/n
is <1, the series converges (absolutely)
is >1, the series diverges
is =1, inconclusive