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Chapter 7: Q. 7 (page 655)

Some Convergent Sequences Involving Exponents: For any real number p > 0, the following sequences converge. Fill in each blank with the appropriate value.

p1k鈫�

Short Answer

Expert verified

The required answer isp1k鈫�1

Step by step solution

01

Step 1. Given information    

The given data is for any real numberp > 0, the sequences converge.

02

Step 2. Explanation    

As we know that the sequence 1kconverges to zero. Since the exponential function pxis continuous for p>0.

Thus, by using limits of function of sequence, which states that if ak鈫�L and f is a function that is continuous at L then f(ak)鈫�f(L)

Thus, the sequence p1kconverges to p0=1

Hence,p1k鈫�1

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Most popular questions from this chapter

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�tan鈭�1鈦�k1+k2

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�鈥�k!kk

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?

Given thata0=-3,a1=5,a2=-4,a3=2and鈭�akk=2鈭�=7, find the value ofrole="math" localid="1648828803227" 鈭�akk=3鈭�.

Determine whether the series 鈭�j=2鈭�2j3converges or diverges. Give the sum of the convergent series.
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