HOMEWORK ASSIGNMENT 1 NOTES DAVID BEN MCREYNOLDS 1 Before We Start When we solve problems one important aspect of problem solving that is often overlooked is believing that you can solve the problem If you do not think that you can do these problems then you need to convince yourself otherwise You must be confident in your ability 2 Section 10 2 This section deals primarily with the question of whether or not a sequence of real numbers is bounded and or monotone Recall that a sequence is bounded if you can find real numbers M1 and M2 such that M 1 an M 2 A sequence is monotone if is satisfies on the the following 1 an an 1 2 an 1 an In the first case we say the sequence an is nondecreasing inequality is strict we say the sequence is increasing In the second case we say the sequence an is nonincreasing inequality is strict we say the sequence is decreasing Lastly we say a sequence an is eventually monotone if there N a positive integer such that the sequence an is monotone n N Date January 20 2002 1 If the If the exists when 2 DAVID BEN MCREYNOLDS 2 1 Problems 1 8 These problems are pattern recognition coupled with testing your ability to construct a formula Example 2 1 Consider the sequence 2 5 10 17 26 We see that a1 2 a2 2 3 a3 2 3 5 a4 2 3 5 7 a5 2 3 5 7 9 an 2 3 5 2n 1 We will see later that we can write this in a more compact notation by n X an 2 2i 1 i 2 2 2 Problems 9 40 These problems comprise the bulk of the first assignment We work a few examples Example 2 2 Consider the sequence 1 an n Now since the denominator is growing and the numerator is fixed we would conjecture that the numbers in the sequence would get small This tells us that the sequence should be bounded We show this as follows 1 n if n 1 1 1 if n 1 n So we have shown that the sequence is bounded above by 1 As for below since the sequence is positive we have 1 0 n So that we have shown the sequence is bounded HOMEWORK ASSIGNMENT 1 NOTES 3 For monotone we have two possible techniques that we could try We could take derivatives of the function we get by replacing n with x or we can look at ratios Important Any time you see the factorial n YOU SHOULD ALWAYS THINK RATIO So we consider the ratio an 1 an 1 n 1 1 n This equals 1 n n 1 1 Now the reason we think of ratios with the factorial is the following n 1 n 1 n So using this we get 1 n 1 n 1 n 1 n 1 Since n 1 we know that n 1 1 Thus 1 1 n 1 So our sequence is monotone decreasing 2 3 Problems 41 and 42 Here is a hint to these problems Do Problem 41 first Try using the ideas we used in the above example involving the factorial Then try doing problem 42 like problem 41 3 Section 10 3 You need to read section 10 3 if you have not Some important results are the following Theorem 3 1 Every convergent sequence is bounded A very important result follows by taking the contrapositive of the above theorem Theorem 3 2 Every sequence that is unbounded is not convergent That is unbounded sequence are divergent This is quite useful in showing a sequence is not convergent The Pinching Theorem or Squeeze Theorem is important as well In fact you can use this to do almost every problem in 1 36 4 DAVID BEN MCREYNOLDS 3 1 Problems 1 36 Try using the Squeeze Theorem and Theorem 3 2 3 2 Problems 51 58 Try rewriting the sequences in a more familiar form Example 3 1 Problem 51 We are given that a1 1 and 1 an 1 an e Writing out the first few terms a1 1 1 1 a2 1 e e 1 a3 a2 e 1 a4 a3 e an 1 1 e 1 e e2 1 1 1 1 2 3 1 e e e 1 1 1 2 3 4 1 e e e 1 en 1 So that the sequence is really an 1 en 1 We know that 1 en 1 when n 1 Recall that e 2 71818 1 So 1 en 1 1 Also we obviously have 0 So that our sequence is bounded 1 en 1 HOMEWORK ASSIGNMENT 1 NOTES 5 To so that the sequence is monotone consider the ratio of consecutive terms an 1 an 1 en 1 en 1 1 en 1 n e 1 1 en 1 n 1 ee 1 1 1 e So that our sequence is monotone decreasing Thus our sequence converges Remark 1 Notice when we have a sequence of the form 1 cn where c is a real number that looking at ratios works quite well Keep this as a general philosophy for these and later problems Now the question wants us to find the limit If you notice we did a lot of work just to show it was convergent Moral Monotone and bounded does not give you the limit We could have just noticed that the sequence 1 en 1 has constant numerator and growing denominator Thus the sequence converges to zero We could also use the squeeze theorem though I think it is a bit artificial and 0 1 en 1 1 n You would need to verify that en 1 n for all positive integers n Also notice that from the definition of our sequence at the beginning of the problem we can see that it is decreasing since 1 an an 1 1 e 1 an an e 6 DAVID BEN MCREYNOLDS 4 Bonus Problems I would like the bonus to be turned in before the first exam When all the bonus problems are turned in I will send out solutions Bonus An important asymptotic relation for the factorial is the following which is a special case of Stirling s Formula n nn e n 2 n where means n lim n nn e n 2 n 1 Use this to show the sequence n n2n is bounded Does this sequence converge an Bonus Here is an easier question Give an example for each of the following a A sequence of real numbers that is bounded but not convergent b A sequence of real numbers an such that an 1 an infinite number of times but the sequence is not bounded c A sequence of real numbers an such that a2n is a convergent sequence but an is not d A sequence of real numbers an such that bn an 1 an is a convergent sequence but an is not e Sequences of real numbers an and bn such that an and bn are both convergent sequences but an …