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Chapter 7: Q. 46 (page 604)

Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.

tan鈦�k蟺2

Short Answer

Expert verified

Ans: The sequence{ak}=tan鈦�k蟺2is not convergent.

Step by step solution

01

Step 1. Given information.

given,

tan鈦�k蟺2

02

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.   

In the sequence {ak}=tan鈦�k蟺2the general term is ak=tan鈦�k蟺2.

03

Step 3. The general term of the sequence is ak=tan⁡kπ2.

The sequenceak=tan鈦�k蟺2is not monotonic because the sign of tan(k蟺2)varies as k increases. Therefore, the given sequence is not monotonic.

04

Step 4. Now,

The sequence {ak}=tan鈦�k蟺2is bounded below because

鈭�鈭�<ak<鈭�fork>0

The given sequence has no lower and upper bounds, therefore, the sequence is not bounded.

05

Step 5. Thus,

The sequence {ak}=tan鈦�k蟺2is not monotonic and not bounded. Therefore, the sequence {ak}=tan鈦�k蟺2is not convergent.

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Most popular questions from this chapter

Find the values of x for which the series 鈭�K=0鈭�sinxkconverges.

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

鈭�k=2鈭�1k(lnk)2

Determine whether the series 鈭�k=2鈭�-35kconverges or diverges. Give the sum of the convergent series.

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.

鈭�k=2鈭�鈥�1k(ln鈦�k)2

Given that a0=-3,a1=5,a2=-4,a3=2and 鈭�akk=2鈭�=7, find the value of鈭�akk=0鈭�.
See all solutions

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