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Chapter 8: Q. 4 (page 679)

What is a Taylor polynomial for a function f at a point x0?

Short Answer

Expert verified

Pn(x)=鈭�k=0nfk(x0)k!(x-x0)k

Step by step solution

01

Step 1. Given information is:

Wehaveafunctionf(x)

02

Step 2. Finding Taylor Polynomial

Considerthatthefunctionfisafunctionwithaderivativeofordern,thenthetaylorpolynomialatx=x0is,Pn(x)=f(x0)+f'(x0)(x-x0)+f''(x0)2!(x-x0)2+....+fn(x0)n!(x-x0)nThegeneralformoftheTaylorpolynomialofthefunctionfinthecompactformis:Pn(x)=鈭�k=0nfk(x0)k!(x-x0)k

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Most popular questions from this chapter

Find the interval of convergence for power series:鈭�k=0鈭�-1k2k+1x2k+1

Let 鈭�k=0鈭�鈥�akxkbe a power series in x with a radius of convergence p. What is the radius of convergence of the power series 鈭�k=0鈭�鈥�akx鈭�x0k? Make sure you justify your answer.

What is x0if the power series 鈭�k=0鈭�鈥�akx鈭�x0kconverges conditionally at both x=-4and x=8.

Find the interval of convergence for power series:鈭�k=0鈭�k24k+33x+7k

Let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a finite radius of convergence p. Prove that if the series converges absolutely at either 卤p, then the series converges absolutely at the other value as well.

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