# Analysis, Calculus. Difference-eq To Differential eq

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A population of weasels is growing at rate of 3% per year. Let wn be the number of weasels n years from now and suppose that there are currently 350 weasels. (a) Write a difference equation which describes how the population changes from year to year. (b) Solve the difference equation of part (a). If the population growth continues at the rate of 3%, how many weasels will there be 15 years from now? (c) Plot wn versus n for n = 0, 1, 2, . . , 100. (d) How many years will it take for the population to double?

See A History of Mathematics by Carl B. Boyer, Princeton University Press, 1985, page 281). (a) Let fn be the number of pairs of rabbits in the nth month. Explain why f1 = 1 and f2 = 1. (b) Explain why fn+2 = fn+1 + fn for n = 1, 2, 3, . .. (c) Compute fn for n = 3, 4, 5, 6, 7, 8 by hand. (d) Compute fn for n = 1, 2, 3, . . , 100. (e) What is lim fn ? n→∞ (f) Compute rn = fn fn+1 for n = 1, 2, 3, . . , 100. Do you think lim rn exists? If so, what is a good approxn→∞ imation for this limit to five decimal places?

Thus, since m 2 may be made arbitrarily large, the sequence {sn } does not have an upper bound. 15) n→∞ and so the harmonic series does not have a sum. Although the partial sums of the harmonic series diverge to infinity, they grow very slowly. For example, if n = 500, 000, 000, then sn is between 20 and 21. That is, 20 < 1 + 1 1 1 + + ··· + < 21. 2 3 500, 000, 000 Problems 1. Find the sum of each of the following infinite series which has a sum. 3 91 89 (j) 5 n=30 ∞ n−1 (l) n=1 ∞ sin(πn) (n) n=1 3n 7n−1 n π 4 cos(πn) n=1 2.