10.2-part2-solutions

# 10.2-part2-solutions - PART II-10.2 Power Series Do not use...

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1 PART II-10.2 Power Series Do not use a calculator! Do your work for Part II on a separate sheet of paper. Give a concluding statement in mathematical sentence or a complete (non-mathematical) written sentence as appropriate for each part. Staple your work for Part I and Part II together before you turn it in. 1. Goal for this problem : Use power series to approximate with an error less than 0.0001. Begin with a power series representation for a function that we do know: a.) Use this geometric power series to find a power series for using substitution. Write this series in closed form, i.e. expressed in series notation with the nth term. What is its interval of convergence? Why? * (Thinking ahead: ). We want to express as a sum of a power series and find the interval of convergence. To do this using substitution then we need to replace the x in by : * Now we need to determine the interval of convergence for the power series * Observe that is a geometric series where . Thus the series converges absolutely when . arctan(.5) 1 1 - x = x n = 1 + x + x 2 + x 3 + ... + n = 0 x n + ... for x < 1 1 1 + x 2 1 1 + x 2 dx = arctan x + C 1 1 + x 2 1 1 - x - x 2 1 1 + x 2 = 1 1 + - x 2 ( ) = 1 + - x 2 ( ) + - x 2 ( ) 2 + - x 2 ( ) 3 + ... = 1 - x 2 + x 4 - x 6 + ... = -

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10.2-part2-solutions - PART II-10.2 Power Series Do not use...

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