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Chapter 7: Q 48. (page 615)

Determine whether the series 鈭�n=1鈭�-1n+13nconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series 鈭�n=1鈭�-1n+13n converges to 14.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�n=1鈭�-1n+13n.

02

Step 2. Find if the series converges or not.

The index starts with 1, 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 geometric series is 鈭�k=0鈭�crk.

Here, the series 鈭�n=1鈭�-1n+13nhas c=13and r=-13.

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

Since r=-13, it follows that the series localid="1648883874265" 鈭�n=1鈭�-1n+13nconverges.

03

Step 3. Find the value to which the series converges.

If the geometric series 鈭�k=0鈭�crkconverges, it converges to c1-r.

So, the series 鈭�n=1鈭�-1n+13nconverges to 131--13, that is 14.

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Most popular questions from this chapter

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

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

3.454545...

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

鈭�k=2鈭�1k(lnk)2

Prove Theorem 7.25. That is, show that the series 鈭�k=1鈭�akand鈭�k=M鈭�akeither both converge or both diverge. In addition, show that if 鈭�k=M鈭�akconverges to L, then鈭�k=1鈭�akconverges tolocalid="1652718360109" a1+a2+a3+....+aM-1+L.
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