Chapter 10 Part 2 Flashcards
Pn(x) =
f(c) + f’(c)(x-c)
+ f’‘(c)(x-c)^2 / 2! + …
+ f^n(c)(x-c)^n / n!
THE REMAINDER/ERROR OF A TAYLOR POLYNOMIAL
watch multiple videos
Error is less than or equal to the next term
|Rn(x)| =
|f(x) - Pn(x)|, which is less than or equal to |max[f^(n+1)(z) * (x-c)^(n+1) / (n+1)!|
When working with an alternating series, and the terms decrease in magnitude, then a(n+1) < b, when finding
an error less than b
maximum value of cos/sin(x) =
1
if given “on [a, b]”, x will equal b for |Rn(x)| because
it is the maximum value of x
- for max[f^(n+1) (z)], find the n+1 derivative of f(x) and plug in b
a power series converges at its _____
center
if the power series diverges by the ratio test, it only converges at its _____, and R = ____
center; 0
R:
radius of convergence
if the power series converges at an interval [a, b], then R =
b-a / 2
- be sure to check endpoints of the interval
if the power series’ limit is less than 1, it converges ______, and R =
everywhere; ∞
MOST POPULAR POWER SERIES
1/1+x =
1 - x + x^2 - x^3 +…+ (-1)^n * x^n
e^x =
1 + x + x^2 / 2! + x^3 / 3! +…+ x^n / n!
sinx =
x - x^3 / 3! + x^5 / 5!
- x^7 / 7! +…
+ (-1)^n x^(2n+1) / (2n+1)!
cosx = `
1 - x^2 / 2! + x^4 / 4!
- x^6 / 6! +…
+ (-1)^n x^2n / 2n!
ln(x) =
(x-1) - (x-1)^2 / 2 + (x-1)^3 / 3 -…+ (-1)^(n-1) * (x-1)^n / n
All of the parts of the power series that have x to the nth degree count as the nth term
Geometric series
Pn(x) = ar^n
binomial series (k>0)
f(x) = (1+x)^k
f^n(x) = [k - (n-1)] * x^n / n!
you can manipulate complex power series using the ones you know
