121616949-math.281 - and the true value is at most the size...

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10.11 Taylor’s Theorem 267 Exercises For each function, find the Maclaurin series or Taylor series centered at a , and the radius of convergence. 1. cos x 2. e x 3. 1 /x , a = 5 4. ln x , a = 1 5. ln x , a = 2 6. 1 /x 2 , a = 1 7. 1 / 1 - x 8. Find the first four terms of the Maclaurin series for tan x (up to and including the x 3 term). 9. Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for x cos( x 2 ). 10. Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for xe - x . One of the most important uses of infinite series is the potential for using an initial portion of the series for f to approximate f . We have seen, for example, that when we add up the first n terms of an alternating series with decreasing terms that the difference between this
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Unformatted text preview: and the true value is at most the size of the next term. A similar result is true of many Taylor series. THEOREM 10.52 Suppose that f is defined on some open interval I around a and suppose f ( N +1) ( x ) exists on this interval. Then for each x ± = a in I there is a value z between x and a so that f ( x ) = N X n =0 f ( n ) ( a ) n ! ( x-a ) n + f ( N +1) ( z ) ( N + 1)! ( x-a ) N +1 . Proof. The proof requires some cleverness to set up, but then the details are quite elementary. We want to define a function F ( t ). Start with the equation F ( t ) = N X n =0 f ( n ) ( t ) n ! ( x-t ) n + B ( x-t ) N +1 ....
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