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**Unformatted text preview: **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 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...

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