Math, asked by sansonkusare, 1 year ago

State and prove Taylors theorem,

Answers

Answered by atiindumthi
0

Taylor’s Theorem. Let f be an (n + 1) times differentiable function on

an open interval containing the points a and x. Then

f(x) = f(a) + f

0

(a)(x − a) + f

00(a)

2! (x − a)

2 + . . . +

f

(n)

(a)

n!

(x − a)

n + Rn(x)

where

Rn(x) = f

(n+1)(c)

(n + 1)! (x − a)

n+1

for some number c between a and x.

The function Tn defined by

Tn(x) = a0 + a1(x − a) + a2(x − a)

2 + . . . + an(x − a)

n where ar =

f

(r)

(a)

r!

,

is called the Taylor polynomial of degree n of f at a. This can be thought

of as a polynomial which approximates the function f in some interval

containing a. The error in the approximation is given by the remainder

term Rn(x). If we can show Rn(x) → 0 as n → ∞ then we get a sequence

of better and better approximations to f leading to a power series expansion

f(x) = X

n=0

f

(n)

(a)

n!

(x − a)

n

which is known as the Taylor series for f. In general this series will

converge only for certain values of x determined by the radius of

convergence of the power series (see Note 17). When the Taylor polynomials

converge rapidly enough, they can be used to compute approximate

values of the function.

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