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Unformatted text preview: (a) X n =0 (-1) n 2 n 4 n (2 n )! (b) X n =1 1-2 n 4 n 10. Find the interval of convergence of X n =1 (-2) n n ( x + 3) n . 11. Find a power series expansion, centered at 0, for f ( x ) = x 2 + x and its radius of convergence. 12. Isaac Newton showed that (1-x 2 )-1 / 2 = X n =0 (2 n )! 4 n ( n !) 2 x 2 n for-1 < x < 1. (a) Using this formula, nd a power series expansion for arcsin x . (b) Use your power series from part (a) with x = 1 / 2 to nd an innite series whose sum is . 13. Use power series expansions to compute lim x e x 2-1 cos x-1 . 14. (a) Find the third-degree Taylor polynomial for f ( x ) = x 4 / 3 about a = 27. (b) Estimate the maximum error involved in estimating f with the Taylor polynomial you found in part (a) for 25 x 29....
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