Calculus II, Spring 2016

Quiz 2

Mar 17, 2016 Show all details.

1. (20 pts) State (need not prove) the Leibnitz test for alternating series. Then give an example of a_{n} such that ∑_{n} a_{n} converges and ∑_{n} |a_{n}| diverges. Answer. Suppose a_{n} ≥ 0 (3 pts), a_{n} is decreasing (3 pts), and a_{n} → 0 (3 pts). Then ∑_{n}(−1)^{n}a_{n} (3 pts) converges (3 pts).

Let a_{n} = (−1)^{n}/n. (5 pts) Then ∑_{n} a_{n} converges and ∑_{n} |a_{n}| diverges.

2. (20 pts) Give an example of a power series that converges on [0, 2] and diverges elsewhere. Explain. Do the same for (3, 7]. Answer. Consider

∑_{n} \frac{(−1)^{n}}{n} (x − 1)^{2n}. (4pts)

Then

lim_{n→∞} \frac{n|x − 1|^{2(n+1)}}{(n + 1)|x − 1|^{2n}} = |x − 1|^{2} < 1 ⇔ x ∈ (0, 2). (2pts)

For x = 0,

∑_{n} \frac{(−1)^{n}}{n} converges by Leibnitz Test. (2pts)

For x = 2,

∑_{n} \frac{(−1)^{n}}{n} converges for the same reason. (2pts)

Thus, this series converges exactly on [0, 2].

Consider

∑_{n} \frac{(−1)^{n}}{n} (\frac{x − 5}{2})^{n} . (4pts)

Then

lim_{n→∞} \frac{n}{n + 1}|\frac{x − 5}{2}| = |\frac{x − 5}{2}| < 1 ⇔ x ∈ (3, 7). (2pts)

For x = 3,

∑_{n} \frac{1}{n} diverges by P-Test. (2pts)

For x = 7,

∑_{n} \frac{(−1)^{n}}{n} converges by Leibnitz Test. (2pts)

Thus, this series converges exactly on (3, 7].

3. (20 pts) Give the power series representation (centered at a = 0) of \frac{1}{1+x} and find its radius of convergence. Then use it to find the power series representation (centered at a = 0) of ln(1 + x). Answer.

\frac{1}{1 + x} = ∑_{n=0}^{∞}(−1)^{n}x^{n} (5pts), R = 1. (5pts)

ln(1 + x) = ∫_{0}^{x} \frac{1}{1 + t} dt (5pts) = ∫_{0}^{x} ∑_{n=0}^{∞}(−1)^{n}t^{n} dt = ∑_{n=0}^{∞}(−1)^{n} \frac{x^{n+1}}{n + 1}. (5pts)

4. (20 pts) Suppose we know the power series representation of sin x (centered at a = 0), sin x = ∑_{n=0}^{∞} \frac{(−1)^{n}x^{2n+1}}{(2n+1)!} is valid for all x ∈ ℝ. Use it to find the power series representation of cos x (centered at a = 0). Next, assume that tan x = ∑_{n=0}^{∞} c_{n}x^{n} on −R < x < R, find c_{0}, ⋯, c_{5}. Answer.

cos x = (sin x)′ = (∑_{n=0}^{∞} \frac{(−1)^{n}x^{2n+1}}{(2n + 1)!} )′ = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(2n)!} x^{2n}. (8pts)

P_{5}(x) = x + \frac{1}{3}x^{3} + \frac{2}{15}x^{5}.

⇒ c_{0} = 0, c_{1} = 1, c_{2} = 0, c_{3} = \frac{1}{3}, c_{4} = 0, c_{5} = \frac{2}{15}. (12pts)

5. (20 pts) True or False? Explain.

If f(x) = ∑_{n=0}^{∞} a_{n}(x − a)^{n} converges on (a − 1, a + 1), then a_{n} = \frac{f^{(n)}(a)}{n!}. Answer. True. (4 pts)

f^{(n)}(x) = a_{n}n! + a_{n+1}(n + 1) … 2(x − a) + a_{n+2}(n + 2) … 3(x − a)^{2} + … (16pts)

⇒ f^{(n)}(a) = a_{n}n!.