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The Taylor Series

Problems

Some Common Taylor Series

Differentiation and Integration of Power Series

Problem : Compute the Taylor polynomial at 0 of degree 3 for f (x) = sin(e x) .

The first few derivatives are


f'(x) = cos(e x)e x  
f''(x) = cos(e x)e x - sin(e x)e 2x  
f'''(x) = f''(x) - [sin(e x)2e 2x + cos(e x)e 3x]  

Substituting 0 for x yields f (0) = sin(1) , f'(0) = cos(1) , f''(0) = cos(1) - sin(1) , f'''(0) = - 3 sin(1) , so the desired Taylor polynomial is:

sin(1) + cos(1)x + + x 3    

Problem : Approximate log(0.8) using the Taylor polynomial of degree 3 at x = 1 for log(x) .

The desired Taylor polynomial is

(x - 1) - +    

Substituting 0.8 for x , we have log(0.8) -0.222 .

Problem : Find the Taylor series for sin(x) + cos(x) .

Adding term-by-term, we have

1 + x - - + + ...    

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