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Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.

If r=1thenrk1

Short Answer

Expert verified

Hence proved,r=1thenrk1

Step by step solution

01

Step 1. Given information

The given sequence Ifr=1thenrk1

02

Step 2. To prove that the result,observe the behavior of geometric sequence forr=1 

The geometric sequence rkwith ratio r=1is a constant sequence with each term equal to 1

The term of the sequence rkis

rk=1,1,1.......

The sequence rkis a constant sequence and is bounded

The constant sequence is always convergent and the sequence rkis converging to 1

Therefore,r=1then rk1holds

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