10.3-10.4

Question 1

What’re principles of the p-series?

Answer 1

• Form: (1/n^p)
• p > 1, converge
• p ≤ 1, diverge

Question 2

What is the Comparison test?

Answer 2

• Take high degree on top and bottom and compare to original
• “Larger converges, so does smaller”
• “Smaller diverges, so does larger”

Question 3

If comparison test doesn’t work, what do you do for L.C.T.?

Answer 3

• MUST be positive terms
• lim->inf (An/Bn) > 0 or else test fails
• if Bn conv/div An will do same

Question 4

For small x:

• sin(x) = x
• ln(x) = x

Answer 4

Ye

Question 5

What is the Remainder Theorem?

Answer 5

*R_n

Question 6

How does Alternating Series Test work?

Answer 6

*abs value. Then prove it decreases and limit to inf is 0. It’ll converge.

Question 7

How to prove absolute/conditional convergence of alt series?

Answer 7

• If both original and abs value converge, it’s abs convergence.
• If original converge but abs diverges, it’s conditionally.

Question 8

What is Remainder Estimate for alternating series?

Answer 8

• R_n ≤ A_n+1 ≤ error

* plug in abs values.

Question 9

What must you do for the Ratio test?

Answer 9

• lim to infinity -> |An+1/An|
• If > 1 or inf diverges
• if 1 fails

Question 10

What needs to be done when performing an integral test?

Answer 10

• positive, continuous, decreasing (Use after T.D. Fails)
• integrate
• “Since integral conv/div, so must series”