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Chapter 7: Q. 5 (page 591)

What is meant by the index of a term of a sequence?

Short Answer

Expert verified

Ans: a0=0,a1=鈭�1,a2=2,a3=鈭�3,鈥�

The subscript of the terms of the sequence is {0,1,2,3,...}.

Hence, the index of the term of the sequence is {0,1,2,3,...}.

Step by step solution

01

Step1. Given Information.

given,

What is meant by the index of a term of a sequence?

02

Step2. The objective is to explain the index of terms of the sequence.

If akis the sequence, then the subscript k is the index of the sequence akwhere k is a positive integer.

Let be the sequence akdefined as (鈭�1)kkk=0鈭�

The terms of the sequence can be found by substitutingk=0,1,2,...

03

Step 3. Therefore,

a0=0,a1=鈭�1,a2=2,a3=鈭�3,鈥�

The subscript of the terms of the sequence islocalid="1649307016389" {0,1,2,3,...}.

Hence, the index of the term of the sequence is {0,1,2,3,...}.

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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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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.

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

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Given a series 鈭�k=1鈭�ak, in general the divergence test is inconclusive when . For a ak鈫�0geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.
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