Mathematics 242 – Principles of Analysis Exam 2 – November 4, 2005

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 Extra Credit Points.

I. A) (20) State and prove the Monotone Convergence Theorem for sequences. (You may

give the proof in the case that the sequence is monotone increasing.) B) (10) Suppose that x_{n} is a sequence of (strictly) negative numbers and x_{n+1}/x_{n} ≤ 1

for all n ≥ 1. Show that lim_{n→∞} x_{n} = a exists in R and satisfies a ≤ 0.

II. A) (10) State the Bolzano-Weierstrass theorem. B) (10) Let x_{n} = sin(n) (the sequence of values of sin(x) at the angles x given by a whole

number n ≥ 1 in radian measure). Show that there exists a subsequence x_{n_{k}} = sin(n_{k}) that converges.

III. Give an example, or give a short proof that there are no such examples. A) (10) A convergent series ∑_{k=1}^{∞} a_{k} such that a_{k} > 0 for all k and lim_{k→∞} ka_{k} = 4. B) (10) A convergent series whose sequence of partial sums is monotone decreasing.

IV. Say whether each of the following infinite series converges absolutely, converges conditionally, or diverges. Justify your answer in each case. A) (10)

∑_{k=1}^{∞} \frac{(−1)^{k+1}}{5k + 3}

B) (10)

∑_{k=0}^{∞} \frac{(−1)^{k}}{7^{k}}

V. (10) Using the Ratio Test and other facts as needed, determine for which x ∈ R the series

∑_{k=0}^{∞} \frac{(x − 3)^{k}}{k6^{k}}

converges.

Extra Credit. (10) Assume that x_{n} is a sequence that converges to a as n → ∞. Construct a new sequence y_{n} from the terms of x_{n} by y_{n} = \frac{x_{1}+x_{2}+···+x_{n}}{n} Show that y_{n} also converges to a as n → ∞. (Hint: it may help to recall that if x_{n} → a, then there is some M such that |x_{n} − a| ≤ M for all n ≥ 1.)