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Chapter 7: Q. 20 (page 624)

Short Answer

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01

Step 1. Given information.

given,

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Most popular questions from this chapter

Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral ∫1∞a(x)dxconverges, then the nth remainder, Rn, for the series∑k=1∞a(k) is bounded by0≤Rn=∑k=n+1∞a(k)≤∫n∞a(x)dx

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

Which p-series converge and which diverge?

Let αbe 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 ∑k=1∞ 12k−1, the sum will be greater than α. Then argue that, by adding in some other finite number of the terms of

∑k=1∞ −12k , 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.)

Given thata0=-3,a1=5,a2=-4,a3=2and∑akk=2∞=7, find the value of∑akk=4∞.
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