1120_14 - Week 14 Outline Binomial Series the formula...

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Week 14 Outline Binomial Series –the formula –examples First-Order Differential Equations –Examples –Separable Equations –First-Order Linear Equations The Binomial Series For - 1 < x < 1, (1 + x ) m = 1 + X k =1 ˆ m k ! x k , where we define ˆ m 1 ! = m, ˆ m 2 ! = m ( m - 1) 2! , and ˆ m k ! = m ( m - 1)( m - 2) · · · ( m - k + 1) k ! for k 3 . Remark If m is a positive integer, then the formula becomes (1 + x ) m = m X k =0 ˆ m k ! x k . In particular, we have m X k =0 ˆ m k ! = 2 m , and m X k =0 ( - 1) k ˆ m k ! = 0 . Examples 1. Expand 1 (1+ x ) 2 as a power series. 2. Find the Taylor series for 1 4 - x about 0 and determine its radius of convergence. 3. Evaluate 3 28 with error at most 0.001. 1
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4. Expand 1 1 - x 2 and use this result to find the Taylor series for sin - 1 x about 0. 5. Expand 1 1+ x 2 and use this result to find the Taylor series for sinh - 1 x about 0. General First-Order Differential Equations and Solutions Definition A first-order differential equation is an equation d y d x = f ( x, y ) , where f ( x, y ) is a function of two variables defined on a region in the xy -plane.
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