121616949-math.262

# 121616949-math.262 - 248 Chapter 10 Sequences and Series...

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248 Chapter 10 Sequences and Series EXAMPLE 10.27 Show that X n =1 5 n diverges. This is almost as easy but the logic has a little twist. If n =1 5 /n 1 / 2 converges then theorem 10.17 implies that n =1 (1 / 5)5 /n 1 / 2 = n =1 1 /n 1 / 2 also converges, but since this is a p -series with p < 1 we know that in fact it does not converge. Therefore the series n =1 5 /n 1 / 2 can’t converge. Since it is typically difficult to compute the value of a series exactly, a good approx- imation is frequently required. In a real sense, a good approximation is only as good as we know it is, that is, while an approximation may in fact be good, it is only valuable in practice if we can guarantee its accuracy to some degree. This guarantee is usually easy to come by for series with decreasing positive terms. EXAMPLE 10.28 Approximate 1 /n 2 to two decimal places. Referring to figure 10.2 , if we approximate the sum by N n =1 1 /n 2 , the error we make is
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