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Chapter 7: Q. 6 (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.

The limit of a sequence.

Short Answer

Expert verified

The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. For example,limn→∞n-1n+1=1

Step by step solution

01

Step 1. Given Information   

The given statement is the limit of a sequence.

02

Step 2. Explanation  

The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity.

The number L is the limit of the sequence anifgivenε>0,an≈Lforn>>1

If such an L exists, we say the sequence is convergent otherwise the sequence is divergent.

For example, limn→∞n-1n+1=1, in this the function is approaching limit equals to 1 as n increases.

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Most popular questions from this chapter

Let a:[1,∞)→ℝbe a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral ∫1∞a(x)dxconverges, then the series localid="1649180069308" ∑k=1∞a(k)does too.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

∑k=2∞ 1kln⁡k

Determine whether the series ∑j=2∞2j3converges or diverges. Give the sum of the convergent series.

What is the contrapositive of the implication “If A, then B"?

Find the contrapositives of the following implications:

If a divides b and b dividesc, then a divides c.

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.

∑k=1∞ 12k−3/4
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