Sequences and Series

284 Chapter 11 Sequences and Seriesseries to represent ln(x) when 0 < x ≤ 2. For exampleand soln(3/2) = ln(1−−1/2) =∞∑(−1) n 1 1n+12 n+1n=0ln(3/2) ≈ 1 2 − 1 8 + 124 − 164 + 1160 − 1384 + 1896 = 9092240 ≈ 0.406.Because this is an alternating series with decreasing terms, we know that the true valueis between 909/2240 and 909/2240 − 1/2048 = 29053/71680 ≈ .4053, so correct to twodecimal places the value is 0.41.What about ln(9/4)? Since 9/4 is larger than 2 we cannot use the series directly, butln(9/4) = ln((3/2) 2 ) = 2ln(3/2) ≈ 0.82,so in fact we get a lot more from this one calculation than first meets the eye. To estimatethe true value accurately we actually need to be a bit more careful. When we multiply bytwo we know that the true value is between 0.8106 and 0.812, so rounded to two decimalplaces the true value is 0.81.Exercises 11.9.1. Find a series representation for ln2. ⇒2. Find a power series representation for 1/(1−x) 2 . ⇒½½º½¼3. Find a powerÌÝÐÓÖËÖ×series representation for 2/(1−x) 3 . ⇒4. Find a power series representation for 1/(1−x) 3 . What is the radius of convergence? ⇒∫5. Find a power series representation for ln(1−x)dx. ⇒We have seen that some functions can be represented as series, which may give valuableinformation about the function. So far, we have seen only those examples that result frommanipulation of our one fundamental example, the geometric series. We would like to startwith a given function and produce a series to represent it, if possible.∞∑Suppose that f(x) = a n x n on some interval of convergence. Then we know thatn=0we can compute derivatives of f by taking derivatives of the terms of the series. Let’s look