Difference Equations to Differential Equations - An by Sloughter D.

By Sloughter D.

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3 that αn−1 + αn−2 + · · · + α2 + α + 1 = Hence xn = αn x0 + β 1 − αn 1−α , 1 − αn . 8) n = 0, 1, 2, . , is the solution of the first-order linear difference equation xn+1 = αxn + β when α = 1. 4 We have seen examples of first-order linear equations in the population growth and radioactive decay examples above. Another interesting example arises in modeling the change in temperature of an object placed in an environment held at some constant temperature, such as a cup of tea cooling to room temperature or a glass of lemonade warming to room temperature.

N→∞ (e) How many years will it take for the population to reach 9500? 3. Do Problem 2 assuming an uninhibited growth model and no restrictions on the number of pike that the lake can support. 4. Suppose rn represents the number of snowshoe rabbits in a certain National Forest in Alaska after n years with an initial value of r0 = 5000. Moreover, suppose the forest can support no more than 10,000 rabbits and {rn }satisfies the inhibited growth model rn+1 = rn + β rn (10, 000 − rn ) 10, 000 for n = 0, 1, 2, .

The resulting infinite series with nth partial sum given by sn = 1 + 1 1 1 + + ··· . 12) is called the harmonic series. 3 The Sum of a Sequence 9 the sequence {sn } is monotone increasing. Hence, by the Monotone Sequence Theorem, {sn } either converges or diverges to infinity. Now s1 = 1, 1 s2 = 1 + , 2 1 s4 = 1 + + 2 1 s8 = 1 + + 2 1 =1+ + 2 1 1 1 1 1 1 1 1 + >1+ + + =1+ + =1+2 , 3 4 2 4 4 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 + + + + + >1+ + + + + + + 3 4 5 6 7 8 2 4 4 8 8 8 8 1 1 1 + =1+3 , 2 2 2 and 16 s16 = s8 + j=9 16 1 1 > s8 + >1+3 j 16 j=9 1 2 + 8 =1+4 16 1 2 .

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