Find the Taylor Series for f(x) = arctan(x) centered at a = 0 in two ways: (a) First, take derivatives of the function to find a pattern and conjecture what the Taylor Series must be. Second, get the same answer by starting with the Taylor Series for 1 which you should know. U 1 1+x² Make a substitution u = -x² to get a Taylor Series for Now take the antiderivative of the result to get the Taylor Series for arctan(x). (Hint: when doing the last step, you should get a "+C" in the antiderivative. But, using what you know about Taylor Series, you should be able to figure out the value of C. Oh, and you should get the same answer as in (a).) 9. Use your answer from the previous problem to give the value of the series 1 1 1 1 1 +... 5 11 1 – 3 + 7 9
Find the Taylor Series for f(x) = arctan(x) centered at a = 0 in two ways: (a) First, take derivatives of the function to find a pattern and conjecture what the Taylor Series must be. Second, get the same answer by starting with the Taylor Series for 1 which you should know. U 1 1+x² Make a substitution u = -x² to get a Taylor Series for Now take the antiderivative of the result to get the Taylor Series for arctan(x). (Hint: when doing the last step, you should get a "+C" in the antiderivative. But, using what you know about Taylor Series, you should be able to figure out the value of C. Oh, and you should get the same answer as in (a).) 9. Use your answer from the previous problem to give the value of the series 1 1 1 1 1 +... 5 11 1 – 3 + 7 9
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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