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Chapter 1: Q1P (page 35)

Prove theorem 14.3. Hint: Group the terms in the error as role="math" localid="1657423688910" (an+1+an+2)+(an+3+an+4)+鈰�to show that the error has the same sign as role="math" localid="1657423950271" an+1.Then group them asrole="math" localid="1657423791335" an+1+(an+2+an+3)+(an+4+an+5)+鈰�to show that the error has magnitude less than|an+1|.

Short Answer

Expert verified

The alternating sequence converges.

S-a1+a2+鈥�+an鈮�an+1

Step by step solution

01

Given Information

The Properties of alternating sequence.

S=鈭�n=0鈭�anan+1<anlimn鈫�鈭�an=0

02

Definition of Alternating series

A mathematical series in which the terms are alternately positive and negative is known as an alternating series.

03

Solve for convergence of given series    

Use properties mentioned below:

S=鈭�n=0鈭�anan+1<anlimn鈫�鈭�an=0

Solve left hand side of the theorem.

S-a1+a2+鈥�+an=an+1+an+2+an+3+an+4鈰�=an+1+an+2+an+3+an+4+鈥�an+1S-a1+a2+鈥�+an=an+1+an+2+an+3+an+4+an+5+鈥�S-a1+a2+鈥�+an鈮�an+1

Hence, the alternating sequence converges.

S-a1+a2+鈥�+an鈮�an+1

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