8.7-8.10 taylor/maclaurin, lagrange, power

Question 1

What is the Lagrange form of the remainder?

Answer 1

|R_n(x)|≤|x-c|ⁿ⁺¹*max of |fⁿ⁺¹(z)|/(n+1)!, where z is a number between x and c, AKA the diff. btw. x and c to the next power multiplied by the max of the next derivative on the interval divided by the next factorial, or the n+1th integral of the rest of the poly.

Question 2

If a function has a radius of convergence > 0…

Answer 2

Then it is differentiable and continuous on the interval (c-R, c+R)

Question 3

How is convergence of a Taylor series determined?

Answer 3

If the lim. as n approaches infinity of the remainder is 0 for all x on an interval, then the taylor series of f converges

Question 4

What is a convergent Taylor series equal to?

Answer 4

The sum of n from 0 to infinity of (x-c)ⁿ*max of fⁿ(c)/(n)!

Question 5

What is the power series for ln(x)?

Answer 5

1+(-1)ⁿ(x-1)ⁿ⁺¹/(n+1) starting at 0

Question 6

What is the power series for e<\sup>x^?

Answer 6

xⁿ/n!

Question 7

What is the power series for sin(x)?

Answer 7

(-1)ⁿ(x)²ⁿ⁺¹/(2n+1)! starting at 0

Question 8

What is the power series for cos(x)?

Answer 8

(-1)ⁿ(x)²ⁿ/(2n)! starting at 0

Question 9

What is the power series for arctan(x)?

Answer 9

(-1)ⁿ(x)²ⁿ⁺¹/(2n+1) starting at 0, or the sin power series without the factorial on the bottom

Question 10

What is the power series for (1+x)^k?

Answer 10

1/0!+kx/1!+k(k-1)x²/2!+k(k-1)(k-2)x³/2!… defined by what k is