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Vector Calculus Independent Study

Unit 9 Sample Test

1. [20 points] Expand ln x in a Taylor series about x = 5, find its radius of convergence, and determine if it converges or diverges on the endpoints of its interval of convergence.

2. [20 points] Calculate

(a)

lim_{n→∞}(−1)^{n} \frac{n}{n + 1}

(b)

lim_{n→∞} \frac{1}{n^{2}} + (0.5)^{n}

(c)

lim_{n→∞} \frac{n}{ln n}

(d)

lim_{n→∞} \frac{2^{n}}{n^{2}}

3. [20 points] Consider the following series:

a = 3 + 3(0.5) + 3(0.5)^{2} + … + 3(0.5)^{n} + …

b = ∑ \frac{n}{2n + 1}

c = ∑ ne^{−n^{2}}

d = 5 + \frac{5}{\sqrt{2}} + \frac{5}{\sqrt{3}} + \frac{5}{\sqrt{4}} + …

e = ∑ \frac{1}{3 + 4^{n}}

f = ∑(−1)^{n} \frac{2n}{3n^{2} − 5}

g = ∑ \frac{n!}{e^{n}}

h = ∑(−1)^{n}3^{1/n}

For each series, determine whether it is convergent or divergent, and explain what test you used.

4. [20 points] Let

f(x) = ∑ \frac{n^{n}}{n!} x^{n}

g(x) = ∑_{n=1}^{∞} \frac{(n + 1)!}{10^{n}}(x − 5)^{n}

For both f and g, find

(a) The radius of convergence, including whether the series converges at the endpoints of the interval of convergence.

(b) The power series representing the improper integral.

(c) The power series representing the derivative.