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Chapter 7: Q 50. (page 615)

Determine whether the series 鈭�n=4鈭�411nconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series 鈭�n=4鈭�411nconverges to 26655.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�n=4鈭�411n.

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�n=4鈭�411nhas c=4114andr=111.

Since r=111, it follows that the series localid="1648886017183" 鈭�n=4鈭�411nconverges.

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=4鈭�411nconverges to 41141-111, that is 41331or equivalently 26655.

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Most popular questions from this chapter

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.

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In Exercises 48鈥�51 find all values of p so that the series converges.

鈭�k=2鈭�1klnkp

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

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,鈭�)such that role="math" localid="1649081384626" limx鈫�鈭�f(x)=伪>0. What can the divergence test tell us about the series 鈭�k=1鈭�f(k)?
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