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Chapter 7: Q. 57 (page 640)

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.

鈭�k=1鈭�k!1路3路5路7路路路(2k-1)

Short Answer

Expert verified

The given series converges.

Step by step solution

01

Step 1. Given Information.  

The given series is鈭�k=1鈭�k!1路3路5路7路路路(2k-1).

02

Step 2. Determine whether the given series converges or diverges.   

To determine whether the series converges or diverges we will use the ratio test since the series has positive terms that meet the hypothesis of the test.

Let the general term isak=k!1路3路5路7路路路(2k-1).

So,ak+1=k+1!1路3路5路7路路路(2k-1)2k+1.

By the ratio test,

蚁=limk鈫�鈭�ak+1ak蚁=limk鈫�鈭�k+1!1路3路5路7路路路(2k-1)2k+1k!1路3路5路7路路路(2k-1)蚁=limk鈫�鈭�k+1!k!2k+1蚁=limk鈫�鈭�k+1k!k!2k+1蚁=limk鈫�鈭�k+12k+1蚁=12

Since role="math" localid="1649165126390" 12<1,the given series converges.

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Most popular questions from this chapter

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak鈫�0, then 鈭�k=1鈭�akconverges.

(b) True or False: If 鈭�k=1鈭�akconverges, then ak鈫�0.

(c) True or False: The improper integral 鈭�1鈭�f(x)dxconverges if and only if the series 鈭�k=1鈭�f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series 鈭�k=1鈭�k-pconverges.

(f) True or False: If f(x)鈫�0as x鈫�鈭�, then 鈭�k=1鈭�f(k) converges.

(g) True or False: If 鈭�k=1鈭�f(k)converges, then f(x)鈫�0as x鈫�鈭�.

(h) True or False: If 鈭�k=1鈭�ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

Determine whether the series 9940-9920+9910-995+鈥�converges or diverges. Give the sum of the convergent series.

Prove that if 鈭�k=1鈭�akconverges to L and 鈭�k=1鈭�bkconverges to M , then the series鈭�k=1鈭�ak+bk=L+M.

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

鈭�k=1鈭�kk2+3

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.
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