Exam 3 Flashcards
(18 cards)
Integrating power series
power rule, leave constants and n as is;
plug in bounds;
evaluate result; may require rewriting
derivative of a power series
list terms, plugging in n = 0, 1, 2…
Ratio Test
if an + 1 / an = some n < 1, the series is absolutely convergent
if an + 1 / an = some n > 1, the series is divergent
if an + 1 / an = 1, the test is inconclusive
Root Test
if n root of |an| = some n < 1, the series is absolutely convergent
if n root of |an| = some n > 1, the series is divergent
if n root of |an| = 1, the test is inconclusive
Alternating Series test
if
- bn is equal or decreasing for all n
- limit of bn = 0
the series is convergent
Taylor Series
c(x - a)^n
where
c = f’n (a) / n!
Power Series
an(x - c)^n
1 / (1 - x) as a series
x^n
R = 1
if a power series has R > 0, it is continuous, and so
f’(x) = c * n(x - a)^(n - 1)
|f(x)dx = c*x - a)^(n + 1) / n + 1
finding radius of convergence
use ratio or root test;
R = 1/L
if L = 0, R is infinite
if L is infinite, R = 0
test endpoints for I
e^x as a series
1/n! * x^n
R is infinite
sin(x) as a series
((-1)^n / (2n + 1)!) x^2n + 1
R is infinite
cos(x) as a series
((-1)^n / (2n)!) x^2n
R is infinite
arctan as a series
((-1)^n / 2n + 1) x^2n + 1;
from 1 / 1 + x^2;
R is infinite
ln(1 + x) as a series
-1^(n + 1) (x^n / n);
R = 1
(1 + x)^k as a series
(k! / n!(k - n!)) x^n;
R = 1
Estimating accuracy - Taylor’s Inequality
|R(x)| <= (M|x - a|^(n + 1)) / (n + 1)!
M - Taylor’s Inequality
(n + 1)th derivative of f(x) <= M
