Chapter 1.8, Problem 69ES

(a) Compute the (approximate) values of the terms in the sequence

${1.01}^{101},{1.001}^{1001},{1.0001}^{10001},{1.00001}^{100001},{1.000001}^{1000001},{1.0000001}^{10000001}\dots$ What number doe these terms appear to be approaching

(b) Use Equation (4) to verify your answer in part (a).

(c) Let $1\le a\le 9$ denote a positive integer. What number is approached more and more closely by the terms in the following sequence?

${1.01}^{a0a},{1.001}^{a00a},{1.0001}^{a000a},{1.00001}^{a0000a},{1.000001}^{a00000a},{1.0000001}^{a000000a}\dots$ (The powers are positive integers that begin and end with the digit $a$ and have $0’\text{s}$ in the remaining positions.)