121616949-math.277 - information about the function So far...

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10.10 Taylor Series 263 What about ln(9 / 4)? Since 9 / 4 is larger than 2 we cannot use the series directly, but ln(9 / 4) = ln((3 / 2) 2 ) = 2 ln(3 / 2) 0 . 82 , so in fact we get a lot more from this one calculation than first meets the eye. To estimate the true value accurately we actually need to be a bit more careful. When we multiply by two we know that the true value is between 0 . 8106 and 0 . 812, so rounded to two decimal places the true value is 0 . 81. Exercises 1. Find a series representation for ln 2. 2. Find a power series representation for 1 / (1 - x ) 2 . 3. Find a power 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 Z ln(1 - x ) dx . We have seen that some functions can be represented as series, which may give valuable
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Unformatted text preview: information about the function. So far, we have seen only those examples that result from manipulation of our one fundamental example, the geometric series. We would like to start with a given function and produce a series to represent it, if possible. Suppose that f ( x ) = ∞ X n =0 a n x n on some interval of convergence. Then we know that we can compute derivatives of f by taking derivatives of the terms of the series. Let’s look at the first few in general: f ± ( x ) = ∞ X n =1 na n x n-1 = a 1 + 2 a 2 x + 3 a 3 x 2 + 4 a 4 x 3 + ··· f ±± ( x ) = ∞ X n =2 n ( n-1) a n x n-2 = 2 a 2 + 3 · 2 a 3 x + 4 · 3 a 4 x 2 + ··· f ±±± ( x ) = ∞ X n =3 n ( n-1)( n-2) a n x n-3 = 3 · 2 a 3 + 4 · 3 · 2 a 4 x + ···...
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