Unit 8 - infinite series

Question 1

power series and determining intervals of convergence for given series
(use ratio test to test limit of n at infinity)

Answer 1

• lim n -> ∞ = 0 (converges for all x, radius = ∞, interval = (-∞, ∞)
• lim n -> ∞ = ∞ (converges when x=c, radius = 0, interval = (c)
• lim n-> ∞ = (1 / radius) | x - c|< 1 ( converges when between -r and r, radius = what |x - c| is less than w/o coefficient in front , interval = (- ratio + c, ratio + c)

Question 2

power series format (one used for given power series, intervals of convergence)

Answer 2

∞
Σ An(x - c)^n
n=0

Question 3

after finding interval of convergence for a power series, ___.

Answer 3

-test endpoints for convergence/divergence and update interval brackets accordingly
- DON’T try ratio test, doesn’t normally work for endpoints (start with divergence test)

Question 4

geometric power series format (terms of making them out of functions)

Note: can use ratio test to determine if converges/diverges

Answer 4

∞ ∞
Σ An(r)^n OR Σ An(r)^(n-1)
n=0 n=1

summation if converges is: a / (1-r)

Question 5

power series and determining intervals of convergence/ratio for developed series
(use what r equals in format and make it < 1 to determine it)

Answer 5

|r| < 1 (converges, |x - c| < r, then radius = what |x -c| is less than)

Question 6

why you don’t need to check the endpoints of a geometric power series: ___.

Answer 6

|r| must be < 1 to converge, so if it equals endpoints, it will diverge

Question 7

common steps for transforming function to make geometric power series:

Answer 7

plug in c value, change whole numbers to keep denominator equivalent, change signs if needed, factor out x’s coefficient, divide everything so it’s (1 - r)