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Q. 3TF

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Found in: Page 617

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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An Improper Integral and Infinite Series: Sketch the function $f (x) = \frac{1}{x}$ for x ≥ 1 together with the graph of the terms of the series $\sum_{k = 1}^{\infty} \frac{1}{k} .$ Argue that for every term $S_{n}$ of the sequence of partial sums for this series, $S_{n} > \int_{1}^{n + 1} \frac{1}{x} d x$ . What does this result tell you about the convergence of the series?

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Let $α$ be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to $α$ . (Hint: Argue that if you add up some finite number of the terms of $\sum_{k = 1}^{\infty} \frac{1}{2 k - 1}$ , the sum will be greater than $α$ . Then argue that, by adding in some other finite number of the terms of

$\sum_{k = 1}^{\infty} (- \frac{1}{2 k})$ , you can get the sum to be less than $α$ . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to $α$ .)

Let $a : [1, \infty) \to ℝ$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral $\int_{1}^{\infty} a (x) d x$ converges, then the series localid="1649180069308" $\sum_{k = 1}^{\infty} a (k)$ does too.

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Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .

(b) A divergent p-series.

Let $\sum_{k = 0}^{\infty} c r^{k}$ and $\sum_{k = 0}^{\infty} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that $\sum_{k = 0}^{\infty} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

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