Calc ii 11.2 Flashcards

What is the Taylor polynomial p_n(x) centered at a?

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

What is p_3(x) for f(0)=4, f'(0)=2, f''(0)=2, f'''(0)=6, centered at 0?

4 + 2x + x^2 + x^3 (write in increasing powers for MyLab)

What check should p_n(x) always pass at the center?

p_n(a) = f(a)

In part (a), find p_2(x). In part (b), evaluate at 1.2. What is the common slip?

Keep x in part (a). Do NOT substitute 1.2 into p_2(x) until part (b).

For f(x) = ln x, a = 1, what is p_2(x)?

(x - 1) - (1/2)(x - 1)^2, and p_2(1.2) = 0.18

What is the Ratio Test limit L?

L = lim as k -> infinity of |a_(k+1) / a_k|

Ratio Test: what if L < 1? What if L > 1? What if L = 1?

L < 1: series converges absolutely. L > 1: series diverges. L = 1: test is inconclusive, use another test.

How do you find the radius R from the Ratio Test?

Solve L < 1 for |x - a|. That gives |x - a| < R, so R is the number on the right.

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 sqrt(x^2)?

|x|, not x. This is why the left endpoint can be missed.

When you solve C|x - a| < 1, what is the interior interval?

|x - a| < 1/C, so (a - 1/C, a + 1/C)

At an endpoint of the interval of convergence, what does the Ratio Test give?

L = 1, so it decides nothing. Test each endpoint separately in the ORIGINAL series.

How do you decide if an endpoint is included in the interval?

Plug that x into the original series and test it on its own. Include it with [ only if that endpoint series converges.

What is the harmonic series, and does it converge or diverge?

Sum 1/k. It diverges.

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

Sum (-1)^k / k. It converges by the Alternating Series Test (terms decrease to 0).

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.

What is the geometric series for 1/(1 - r), and when does it converge?

Sum r^k from k=0, converges when |r| < 1, sum = 1/(1 - r)

What is the power series for f(x) = 1/(1 - 17x) centered at 0?

Sum (17x)^k = sum 17^k * x^k, with R = 1/17 from |17x| < 1

How do you differentiate a power series term by term?

Differentiate x^k to get k*x^(k-1). Constants like 17^k stay. The k=0 term becomes 0 and disappears.

What is the series for g(x) = 17/(1 - 17x)^2, where g = f' and f(x) = 1/(1 - 17x)?

Sum from k=1 to infinity of k * 17^k * x^(k-1)

Does differentiating a power series change its radius R?

No, R stays the same. But the endpoints must be re-tested, because the terms changed.

What is the Taylor series for e^x centered at 0, and its R?

Sum x^k / k!, R = infinity, interval (-infinity, infinity)

What is the Taylor series for sin x centered at 0?

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

What is the Taylor series for cos x centered at 0?

Sum (-1)^k * x^(2k) / (2k)!, R = infinity

What is the Taylor series for 1/(1 - x) centered at 0, and its R?

Sum x^k, R = 1, interval (-1, 1) - both endpoints diverge

What is the Taylor series for ln(1 + x) centered at 0?

Sum (-1)^k * x^(k+1) / (k+1), R = 1

Why can't you judge an endpoint series from its first term only?

Terms can start at -5 or 5 and still not go to 0, or can look small at first and diverge. Write out several terms and check the limit of the terms.

If the terms of a series do not go to 0, what can you conclude?

The series diverges, by the Divergence Test.