By Richard. Abbatt

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**Example text**

100. 7. A radioactive element is known to decay at the rate of 2% every 20 years. (a) If initially you had 165 grams of this element, how much would you have in 60 years? (b) What is the half-life of this element? (c) Suppose that the bones of a certain animal maintain a constant level of this element while the animal is living, but the element begins to decay as soon as the animal dies. If a bone of this animal is found and is determined to have only 10% of its original level of this element, how old is the bone?

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. As another example, suppose that an electronic switch works with probability p and fails with probability q = 1 − p. Then, using reasoning analogous to that used in the coin tossing example, the probability that the first failure will occur on the nth use of the switch is pn−1 q, n = 1, 2, 3, . .. (a) Can you justify this probability?

If r = 1, sn = nc and so {sn } does not converge. If r = 1, it is easy to see, using long division (or the derivation in outlined in Problem 4), that 1 − rn = 1 + r + r2 + · · · + rn−1 . 8) Hence, if r = 1, c(1 − rn ) . 9), it is clear that {sn } does not converge if |r| ≥ 1. But if −1 < r < 1, then sn = lim rn−1 = 0, n→∞ and so the sequence has the sum c(1 − rn ) c = . 10) That is, we have now seen that ∞ crn−1 = n=1 whenever −1 < r < 1. 11) with c = 1 and r = 12 , ∞ n=1 1 2 n−1 1 1 1 1 + + + + ··· = 2 4 8 16 =1+ 1 1 1− 2 = 2.

### A treatise on the calculus of variations by Richard. Abbatt

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