Thus we see inductively that a k 1 a k 0 for all k

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Thus, we see inductively that a k +1 - a k > 0 for all k . Therefore the sequence is monotone (increasing). Let us see that if it is bounded.We do this again using induction. First note that a 1 < 6. Next, assume that a k < 6. Then a k +1 = 6 + a k < 6 + 6 < 36 = 6. Thus we have shown inductively that the sequence is bounded by 6. Hence, by BMCT, it converges, to say L . Of course L > 0 since the sequence is increasing and consists of positive numbers. How to find the limit? Consider the equation a k = 6 + a k - 1 . Taking the limit as k → ∞ we get lim k →∞ a k = lim k →∞ 6 + a k - 1 L = 6 + L or L 2 = 6 + L or L 2 - L - 6 4
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or ( L + 2)( L - 3) = 0 Thus L is - 2 or 3. We must rule out the negative number since L is positive. Thus L = 3. Note: The argument given in the above paragraph requires that we already know that the limit exists! 2.6 Example Show that the sequence given recursively by a 1 = 1, a k +1 = a k + 1 k ! is convergent. SOLUTION: The sequence is a k = 1 + 1 1! + 1 2! + · · · + 1 ( k - 1)! . Since a k +1 = a k + 1 k ! and a k > 0 , it is clear that a k +1 a k so we need only show that the sequence is bounded to apply BMCT. For this we note that k ! = 1 · 2 · 3 · 4 · · · · · · k k - 1 factors 1 · 2 · 2 · 2 · · · · 2 k - 1 factors = 2 k - 1 Thus 1 k !
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