11.3 Taylor and Maclorena series

What is the Taylor series of f(x) centered at a?

Sum from n=0 to infinity of [f^(n)(a) / n!] * (x - a)^n

What is a Maclaurin series?

A Taylor series centered at a = 0: sum [f^(n)(0) / n!] * x^n

What is the difference between a Taylor polynomial and a Taylor series?

A Taylor polynomial p_n(x) is finite, up to n terms. A Taylor series is infinite, sum to infinity.

How do you find the first few terms of a Taylor series from scratch?

Find f(a), f'(a), f''(a), f'''(a)... then use f(a) + f'(a)(x-a) + f''(a)/2! * (x-a)^2 + f'''(a)/3! * (x-a)^3 + ...

What is the Maclaurin series for e^x, and its interval?

Sum x^n / n! = 1 + x + x^2/2! + x^3/3! + ... , R = infinity, (-infinity, infinity)

What is the Maclaurin series for sin x?

Sum (-1)^n * x^(2n+1) / (2n+1)! = x - x^3/3! + x^5/5! - ... , R = infinity

What is the Maclaurin series for cos x?

Sum (-1)^n * x^(2n) / (2n)! = 1 - x^2/2! + x^4/4! - ... , R = infinity

What is the Maclaurin series for 1/(1-x), and its interval?

Sum x^n = 1 + x + x^2 + x^3 + ... , for |x| < 1, interval (-1, 1)

What is the Maclaurin series for ln(1+x), and its interval?

Sum (-1)^n * x^(n+1) / (n+1) = x - x^2/2 + x^3/3 - ... , R = 1, interval (-1, 1]

What is the Maclaurin series for 1/(1+x), and its interval?

Sum (-1)^n * x^n = 1 - x + x^2 - x^3 + ... , for |x| < 1

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

Sum [k choose n] * x^n = 1 + kx + k(k-1)/2! * x^2 + ... , for |x| < 1

How do you get the Maclaurin series for e^(2x) from e^x?

Substitute 2x for x: sum (2x)^n / n! = sum 2^n * x^n / n!

How do you get the Maclaurin series for e^(-x^2) from e^x?

Substitute -x^2 for x: sum (-x^2)^n / n! = sum (-1)^n * x^(2n) / n!

How do you get the Maclaurin series for sin(2x) from sin x?

Substitute 2x for x: sum (-1)^n * (2x)^(2n+1) / (2n+1)!

How do you get the Maclaurin series for cos(x^2) from cos x?

Substitute x^2 for x: sum (-1)^n * (x^2)^(2n) / (2n)! = sum (-1)^n * x^(4n) / (2n)!

How do you get the series for 1/(1-x)^2 from 1/(1-x)?

Differentiate 1/(1-x) = sum x^n term by term: 1/(1-x)^2 = sum n * x^(n-1), starting at n=1

How do you get the series for ln(1+x) from 1/(1+x)?

Integrate 1/(1+x) = sum (-1)^n * x^n term by term: ln(1+x) = sum (-1)^n * x^(n+1) / (n+1)

How do you get the series for sin x from cos x?

Integrate cos x = sum (-1)^n * x^(2n) / (2n)! term by term: sin x = sum (-1)^n * x^(2n+1) / (2n+1)!

What is the Taylor series for ln x centered at a = 1?

Sum (-1)^(n-1) * (x-1)^n / n, starting at n=1: (x-1) - (x-1)^2/2 + (x-1)^3/3 - ...

What is the Taylor series for 1/x centered at a = 1?

Sum (-1)^n * (x-1)^n = 1 - (x-1) + (x-1)^2 - (x-1)^3 + ... , for |x-1| < 1

What is the Taylor series for f(x) = 6cos(2x) centered at a = pi/12, up to p4?

p4(x) = 3sqrt(3) - 6(x-pi/12) - 6sqrt(3)(x-pi/12)^2 + 4(x-pi/12)^3 + 2sqrt(3)(x-pi/12)^4

For f(x) = ln x, a = 1, what is p2(x)?

(x-1) - (1/2)(x-1)^2, and p2(1.2) = 0.18. Keep x in part (a), substitute only in part (b).

When you substitute into a known series, what happens to the interval of convergence?

Substitute into the condition |...| < 1 and solve for x. The radius can change: e.g. e^(2x) still R = infinity, but 1/(1-2x) has R = 1/2.

How do you find the radius of a Taylor series you just built?

Use the Ratio Test or Root Test on the new series, then solve the limit < 1 for |x-a|.

What is the limit of k/(k+1) as k -> infinity?

1. Check: 0.9 at k=9, 0.99 at k=99, 0.999 at k=999.

What is the limit of (k+1)/k^2 as k -> infinity?

0. Check: 0.12 at k=9, 0.01 at k=99, 0.001 at k=999. A value at one k is NOT the limit.

What is the limit of (17k+17)/k as k -> infinity?

17, not infinity. (17k+17)/k = 17 + 17/k, and 17/k -> 0.

What is sqrt(x^2)?

|x|, not x. Use it when solving x^2 < 6 to get |x| < sqrt(6).

At an endpoint of an interval of convergence, what must you do?

Test that endpoint separately in the ORIGINAL series. The Ratio/Root limit is 1 there and decides nothing.

How do you decide if an endpoint gets [ or ( in interval notation?

[ if the endpoint series converges, ( if it diverges. Brackets only where it converges.

What is the harmonic series, and does it converge?

Sum 1/k, p-series with p = 1, diverges.

What is the alternating harmonic series, and does it converge?

Sum (-1)^k/k, converges conditionally by the Alternating Series Test (1/k decreases to 0). The p-series test is for positive terms only.

What are the two conditions for the Alternating Series Test?

1) Terms decrease in absolute value. 2) Terms go to 0.

When does a p-series sum 1/k^p converge?

Converges if p > 1, diverges if p <= 1.