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Chapter 7: Q. 7 (page 655)

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

A series, including the meaning of a term of the series

Short Answer

Expert verified

A series is defined as the sum of the elements of a sequence. For example, 2+4+6+8 is the series.It is a sum of elements that follow a pattern. Sum of infinite terms in a series is possible in some cases as well.

Step by step solution

01

Step 1. Given Information    

The given statement is a series, including the meaning of a term of the series

02

Step 2. Explanation   

A series is defined as the sum of the elements of a sequence.

In a finite series, a finite number of terms are written like a1+a2+a3+....an.In case of an infinite series, the number of elements are not finite i.e. a1+a2+a3+....an+....

For example, 2+4+6+8is a series , where the sum of the series or value of the series will be 20.

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Most popular questions from this chapter

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

鈭�k=1鈭�k3k+100

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

鈭�k=1鈭�鈥�k鈭�3/2

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak鈫�0, then 鈭�k=1鈭�akconverges.

(b) True or False: If 鈭�k=1鈭�akconverges, then ak鈫�0.

(c) True or False: The improper integral 鈭�1鈭�f(x)dxconverges if and only if the series 鈭�k=1鈭�f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series 鈭�k=1鈭�k-pconverges.

(f) True or False: If f(x)鈫�0as x鈫�鈭�, then 鈭�k=1鈭�f(k) converges.

(g) True or False: If 鈭�k=1鈭�f(k)converges, then f(x)鈫�0as x鈫�鈭�.

(h) True or False: If 鈭�k=1鈭�ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

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...

Let鈭�k=0鈭�crkand鈭�k=0鈭�bvk be two convergent geometric series. If b and v are both nonzero, prove that 鈭�k=0鈭�crkbvk is a geometric series. What condition(s) must be met for this series to converge?
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