Mathematics 242 – Principles of Analysis Problem Set 5, Due: 10/14

I. Decide whether or not each of the following infinite series converges, and justify your claim using one of the results in sections 2.4 and 2.7 of Abbott:

A)

∑_{n=0}^{∞} \frac{1}{\sqrt{2n + 1}}

B)

∑_{n=0}^{∞} \frac{1}{(2n + 1)^{2}}

C)

∑_{n=0}^{∞}(−1)^{n} \frac{2^{n}}{3^{2n}}

D)

∑_{n=0}^{∞}(−1)^{n} \frac{n^{3} + n}{2n^{3} + n + 1}

II. A) For which x ∈ [0, 2π] does the series

∑_{n=1}^{∞} sin^{n}(x)

converge? What is the sum for those x? B) For which x ∈ R does the series

∑_{n=1}^{∞} e^{nx}

converge? What is the sum for those x?

III. From Abbott: 2.7.3, 2.7.4, 2.7.9.

IV. Use the Ratio Test (Exercise 2.7.9 above) to determine for which x ∈ R the series

∑_{n=0}^{∞} \frac{(x − 1)^{n}}{3^{n}}

converges. (Hint: The Ratio Test will give an open interval of values of x on which the series converges absolutely. What happens at the endpoints of that interval?)