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Chapter 7: Q 49. (page 615)

Determine whether the series 鈭�j=2鈭�2j3converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series鈭�j=2鈭�2j3diverges.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�j=2鈭�2j3.

02

Step 2. Find if the series converges or not.

The index starts with 2, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of a geometric series is 鈭�k=0鈭�crk.

The geometric series converges if and only if r<1.

Here, the series 鈭�j=2鈭�2j3has c=13and r=2.

Since r=2, it follows that the series 鈭�j=2鈭�2j3diverges.

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Most popular questions from this chapter

Improper Integrals: Determine whether the following improper integrals converge or diverge.

鈭�1鈭�1xdx.

Prove Theorem 7.24 (a). That is, show that if c is a real number and鈭�k=1鈭�ak is a convergent series, then 鈭�k=1鈭�cak=c鈭�k=1鈭�ak.

Let a:[1,鈭�)鈫�鈩�be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral 鈭�1鈭�a(x)dxconverges, then the series localid="1649180069308" 鈭�k=1鈭�a(k)does too.

Determine whether the series 鈭�k=0鈭�e蟺3kconverges or diverges. Give the sum of the convergent series.

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
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