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Chapter 7: Q. 66 (page 605)

Complete the proof of Theorem 7.18 by evaluating the limits

of the sequences.

Prove that 1kp鈫�0, when p>0.

Short Answer

Expert verified

The theorem is hence proved.

Step by step solution

01

Step 1. Given Information.

The objective is to prove that1kp鈫�0.

02

Step 2. The proof of the theorem.

The sequence 1kis a decreasing sequence as kincreases, 1kdecreases.

For p>0, for increasing k, the terms of the sequence 1kpdecreases. The sequence 1kpis a decreasing sequence.

It is given that p>0, therefore, limk鈫�鈭�1kp=0.

For 蔚>0,

1kp-0<蔚

There exists an Ngreater than 1蔚1pthen,

1kp-0<蔚for k鈮�N

Thus, 1kpis a null sequence.

Hence,limk鈫�鈭�1kp=0.

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Most popular questions from this chapter

Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series 鈭�k=1鈭�akfor convergence.

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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�1kk
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