B. Let r > 0 be a positive real number. This problem will give a simple approach to lim,-o ". i. Explain briefly why a, = r" can be written recursively as ao = 1, a, = r · an-1 for n > 0. ii. Using...

0. ii. Using the technique of solving for the limit of a recursive sequence, find the possible limits of a, if the sequence converges. iii. Show that a, is monotone. (It may be increasing or decreasing, depending on r.) iv. Combine (ii) and (iii) to show lim,-o " converges to 0 for r < 1,="" converges="" to="" 1="" for="" r="1," and="" diverges="" to="" o="" for="" r=""> 1. "/>
Extracted text: B. Let r > 0 be a positive real number. This problem will give a simple approach to lim,-o ". i. Explain briefly why a, = r" can be written recursively as ao = 1, a, = r · an-1 for n > 0. ii. Using the technique of solving for the limit of a recursive sequence, find the possible limits of a, if the sequence converges. iii. Show that a, is monotone. (It may be increasing or decreasing, depending on r.) iv. Combine (ii) and (iii) to show lim,-o " converges to 0 for r < 1,="" converges="" to="" 1="" for="" r="1," and="" diverges="" to="" o="" for="" r=""> 1.