Chapter 11 Review (Test 2)

Question 1

Ratio Test for Absolute Convergence

Answer 1

p = lim of k approaches infinity of absolute value of a sub k +1 divided by a sub k
If p > 1 , converges for x=0
If p < 1, converges for all x
If p involves x, converges for (a-R, a+R)

Question 2

P-Series Test

Answer 2

If p>1, then converges, alternating converges

If 0 < p less than or equal to 1 then diverges, alternating converges

Question 3

Taylor/Maclaurin Polynomials

Answer 3

sigma of k = 0 to infinity of f to the k derivative of a over k factorial times (x-a) to the k

Question 4

Maclaurin series
E to the x
and E to the -x

Answer 4

1 + x + (x^2)/2! + (x^3)/3! +….

1 - x + (x^2)/2! - (x^3)/3! +….

Question 5

Maclaurin series

sinx

Answer 5

x - (x^3/3!) + (x^5/5!) - …

Question 6

Maclaurin series

cosx

Answer 6

1 - (x^2)/2! + (x^4)/4! - …

Question 7

Maclaurin series

1 / 1-x
1 / 1+ x

Answer 7

1 + x + x^2 + x^3 + …

1 - x + x^2 - x^3

Question 8

Maclaurin series

ln(1+x)

Answer 8

x - (x^2/2) + (x^3)/3 - …

Question 9

Maclaurin series

inverse tan

Answer 9

x - (x^3/3) + (x^5/5) - ….

Question 10

Error

Answer 10

If alternating, convergent series,

absolute value of error is less than or equal to absolute value of next term

Question 11

Langrange’s Form of Remainder

Answer 11

If not alternating and convergent;
Absolute value of R sub n of x is less than or equal to the absolute value of M over (n+1)! times (x-a)to the n+!

M = max value of f to the n+1 of x