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Geometric Series and p-series:

Suppose r is a nonzero real number and p > 0. Fill in the blanks.

For p______ , the p-series k=01kpconverges and for p______, the series diverges.

Short Answer

Expert verified

The required answer is for p>1 , the p-series k=01kpconverges and for p<1, the series diverges.

Step by step solution

01

Step 1. Given Information  

The given data isr is a nonzero real number andp > 0

02

Step 2. Explanation  

By using the concept of convergence of a geometric series,

Suppose r is a non zero real number. Then,

For p>1 , the p-series k=01kpconverges and for p<1, the series diverges.

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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 ak0, then k=1akconverges.

(b) True or False: If k=1akconverges, then ak0.

(c) True or False: The improper integral 1f(x)dxconverges if and only if the series k=1f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series k=1k-pconverges.

(f) True or False: If f(x)0as x, then k=1f(k) converges.

(g) True or False: If k=1f(k)converges, then f(x)0as x.

(h) True or False: If k=1ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

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