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1 \\/1+;c' (a) Find rst four terms of the Maclaurin series for f (9:). 3. Let f(a:) = (b) Use your result to nd the Maclaurin series for f (9:) on k (c) For the power series representation of f(:l:), ns) = Z cnz\EXAMPLE 13 Find the first three nonzero terms in the Maclaurin series for (a) e* sin x and (b) tan x. SOLUTION (a) Using the Maclaurin series for e* and sin x in Table 1, we have e* sin x = + X 1! 2! 3! x 3! + ...) We multiply these expressions, collecting like terms just as for polynomials: 1 + x+ 5x2 + gx3+ ... X X ox + ... x + x2+ 5x3 + 1x4+ ... + - 6x3 - 1x4 -.. x + x2+ zx3+ ... Thus ex sin x = x+ x2 +-x 3x3 +... (b) Using the Maclaurin series in Table 1, we have X 31 51 tan x = sin x COS X 1 2! 41 We use a procedure like long division: 2 x5+.. 1 - 2x2 + 24x4 -... ) x- 1x3+ 120 X's - . . x - 3x3 + 24 J X5 -... 5x3 - 30 x5 + ... 3x3 - 1x5+... 15 x5 + ... Thus tan x = x tax'tix' +... Although we have not attempted to justify the formal manipulations used in Exam- ple 13, they are legitimate. There is a theorem which states that if both f(x) = > Cnx" and g(x) = > bax" converge for |x|