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Chapter 7: Q. 3 (page 624)

What is meant by a p-series?

Short Answer

Expert verified

The series of the type 鈭�k=1鈭�1kp is called p-series.

Step by step solution

01

Step 1. Given Information.

p-series.

02

Step 2. p-series.

The series of the type 鈭�k=1鈭�1kp are the p-series.

03

Step 3. Convergent of divergent.

The p-test states that:

(i) For p>1, the series 鈭�k=1鈭�1kpconverges.

(ii) For p=1, the harmonic series 鈭�k=1鈭�1kdiverges.

(iii) For p<1, the series 鈭�k=1鈭�1kp diverges.

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Most popular questions from this chapter

If limk鈫�鈭�akbkWhereLis a positive finite number, what may we conclude about the two series?

Determine whether the series 鈭�k=0鈭�蟺e3kconverges or diverges. Give the sum of the convergent series.

Whenever a certain ball is dropped, it always rebounds to a height60% of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of 1 meter?

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.

Let 鈭�k=1鈭�akbeaconvergentseriesand鈭�k=1鈭�bkbeadivergentseries.Prove that the series 鈭�k=1鈭�ak+bkdiverges.
See all solutions

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