Please solve these questions. Thank you!

Please answer the following questions in your own words and in the simplest possible terms. As long as you write an answer you will be granted the marks for these questions. Your answers should reflect what you understand and do not have to be correct for you to receive the marks. 1. Give an example for each of the following: (a) a sequence that is not eventually monotonic but converges (b) a decreasing sequence without a limit (c) a bounded sequence without a limit 2. In this question, {a,}" is a sequence of real numbers: (a) Let / be a positive integer. What is the difference between the following? i. lim an N n-+0C iii. M an n=1 N ii. lim E an iv. an N+00 n=1 n=1(b) If the sequence {an Joe, converges, then does the sequence {(-1)"ane, converge too? Explain. 3. Consider the following two series: 18 SI = 2 2V/2 DO (4n)! (1103 + 26390n) S2 = 9801 (n!)43964n n= (4n + 1) (4n + 3) n=0 It turns out that S1 = #/4 (a result from around 1676 attributed to Gottfried Leibniz) and $2 = 1/* (a result from around 1910 attributed to Srinivasa Ramanujan). Let us suppose that we want to estimate the value of 7. How many correct decimal places of # do you get by using (a) one and (b) two terms of each of these series (you can try higher number of terms using e.g., Wolfram Alpha)? Based on your observations, which of these two series would you choose to approximate * and why? Note: you should be able to show that these two series actually converge (no need to show your work for this). 4. What is wrong with the following calculation? 0 = 0+0+0+... = (1- 1) + (1- 1) +(1 -1) + . .. =1+(-1+1)+(-1+1)+(-1+1) ... =1+0+0+0+ ... =1