calc 2

MATH 150 B EXAM #2 Spring 2018

Name: . . . . . . . . . . . . . . . . . . . . . . . .

Seat number: . . . . . . . . . . . . . . . . . . . . . . . . Each problem has 8 points except problem 3 that has 12 points.

1. Find the sum of the following series: a)

∑∞ n=2

1 n(n−1)

b) ∑∞

n=1 5n−1

6n

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2. A) Determine whether each of the following infinite series converges or diverges. (State the test(s) that you have used and show your work). a)

∑∞ n=1

3+8 sin n n2+1

.

b) ∑∞

n=1 n4

n4+7 .

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c) ∑∞

n=1 5+2n

1+2n2+n4

B) Determine if the following series converge absolutely, converge conditionally or diverge Consider the series∑∞

n=1(−1) n+1 3√

5n

3

3. Find the interval of convergence and the radius of convergence for the following power series. (don’t forget to check the endpoints)

∑∞ n=1 (−1)

n−1n− 1 2 (4x2)n.

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4. Recall that 1 1−x =

∑∞ n=0 x

n for |x| < 1. Find a power series representation for the following functions and state the radius of convergence for the power series f(x) = x

2

1+x2 .

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5. The Taylor polynomial T4(x) of degree 4 for (9 + x) 1/2 is

(9 + x)1/2 ≈ 3 + 1 6 x− 1

216 x2 + 1

3888 x3 − 5

279936 x4.

a) Given the above polynomial find the Taylor polynomial for (9 + x)3/2 at a=0. The Taylor polynomial is the first few terms of the Taylor’s series.

b) Use the Taylor polynomial for (9 + x)3/2 obtained in part (a) to approximate √

10 3/2

. No need to simplify the arithmetic.

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6. a) Approximate the function f(x) = x2/3 by a Taylor polynomial of degree 2 at a = 8

b) What is the maximum error when 7 ≤ x ≤ 9?

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