Exam #3 Review

Question 1

Indeterminate forms

Answer 1

∞/∞, 0/0, -∞/+∞, 0^0, 1^∞, ∞^∞, ∞^0, 0*∞

Question 2

L’Hopital (indeterminate fractions)

Answer 2

limx->a f(x)/g(x) = ∞/∞, 0/0, -∞/+∞

Limx->a f’(x)/g’(x) = L (repeat if still indeterminate or use appropriate method.)

Question 3

Indeterminate powers

Answer 3

limx->a [f(x)]^g(x) = 0^0, 1^∞, ∞^∞, ∞^0

Limx->a e^(ln{[f(x)]^g(x)}) =e^L

Question 4

Indeterminate products

Answer 4

limx->a f(x)g(x) = ∞^0, 0∞
Limx->a f(x)/[1/g(x)] OR g(x)/[1/f(x)] which ever is easier
Idea is to write in a form of ∞/∞, 0/0, -∞/+∞ to use L’Hopital’s rule.

Question 5

Improper integrals

Answer 5

Integral from -∞ to +∞, integral from -∞ to #, or integral from # to +∞.
Replace ∞ with a dummy variable (t) and write as a limit as t goes to ∞
L is finite => integral is convergent
L is infinite => integral is divergent
For Integral from -∞ to +∞, split into two integrals: integral from -∞ to # (pick any) + integral from # to +∞
Both must be convergent in order for the integral to converge.

Question 6

Singularities within bounds

Answer 6

A vertical asymptote occurs with in the bounds of the integral. The limit from both sides must be evaluated. Both sides must converge, so if one is divergent the whole thing is divergent.

Question 7

Comparison theorem

Answer 7

Use when the limit of the sequence is in an indeterminate form. If f(x) exists and f(n)=An, then if the limx->∞f(x)=L, the limn->∞An=L too.

Question 8

Absolute theorem

Answer 8

If limn->∞|An|=0, then the limn->∞An=0 also

Question 9

Strategies for Sequences

Answer 9

1. evaluate the limit of An as n approaches ∞
2. Comparison theorem if that is indeterminate
3. If oscillating terms, then use absolute theorem.

Question 10

Absolute convergence and conditional convergence

Answer 10

If the series |An| converges, then series An converges absolutely.
If the series |An| diverges, but the series An still converges, then series An converges conditionally.

Question 11

Power series

Answer 11

Series Cn(x-a)^n ALWAYS USE RATIO TEST!!!
Given a power series, for what values of x will it converge?
If a constant limit is found, the series doesn’t depend on x; thus, the radius of convergence is infinite.
If the limit is going to infinity, then the series only converges at x=a; thus, the radius of convergence is zero.
If the limit depends on |x-a|, then the limit should be set to less than one; thus, the radius of convergence is |x-a|