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**Extra resources for Analysis, Calculus. Difference-eq To Differential eq **

**Example text**

Note that it appears that lim xn = 500, n→∞ while lim yn = ∞. 7) for n = 0, 1, 2, . .. Thus if 0 < β < 1, and we start with x0 < M , then xn < M for all n. Moreover, since xn β > 0, M we have xn+1 = xn + β xn (M − xn ) > xn M for all n. Hence the sequence {xn } is monotone increasing and bounded, and so must have a limit. In Problem 8 you will be asked to verify that this limit is in fact M , as appeared to be the case in the previous example. If β > 1, it may be the case that there are values of n for which xn > M , in which case β xn (M − xn ) < 0 M and, as a consequence, xn+1 < xn .

A) Find the probability P of a female’s reproducing before dying. (b) Plot P as a function of r for 0 ≤ r ≤ 1. 5. 11. How many terms of the harmonic series are needed to obtain a partial sum larger than 5? How many terms are needed to obtain a partial sum larger than 10? 12. Plot the points (n, sn ), where sn is the nth partial sum of the harmonic series, for n = 1, 2, 3, . . , 1000. What does this show you about the rate of growth of the partial sums? 13. The first example of this section is a particular case of the more general problem of computing probabilities associated with the waiting time for some event to occur.

02xn , n = 0, 1, 2, . , makes xn a function of n. For example, the size of a certain population of owls will be a function of the number of years from some starting date. Example The area of a circle is a function of the radius of the circle. Example The distance of the earth from the sun is a function of the time of year. 1 The temperature at a certain fixed point in space is a function of time. In mathematical terminology, if y is a function of x, then we call x the independent variable and y the dependent variable.