Calculus II, Spring 2016

Quiz 2

Mar 17, 2016 Show all details.

1. 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.

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

3. 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).

4. 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). Then use them to find the first few terms of Taylor series generated by tan x (centered at a = 0), upto and including the coefficient of x^{5}.

5. True or False? Explain.

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

Calculus II, Spring 2016

Quiz 2

Mar 17, 2016 Show all details.

1. 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.

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

3. 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).

4. 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). Then use them to find the first few terms of Taylor series generated by tan x (centered at a = 0), upto and including the coefficient of x^{5}.

5. True or False? Explain.

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