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Taylor Series Calculator
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In a Taylor Series, the Taylor degree of a polynomial is the function that can be defined from the function of f. For the Taylor approximation of the function have a polynomial of degree. It can be differentiate by n times.

Taylor Series Calculator (or Taylor Polynomial Calculator) is a tool which calculates Taylor series for a polynomial.

Below is given a default function with its degree of polynomial, click "Submit". The calculator differentiate the function n times and by using Taylor series formula, it will substitute the values of differentials. Then, represent the Taylor series as sum of its terms.

## Steps For Taylor Series

Step 1 :

If f(x) is continuous, and has continuous derivatives through order n-1 on the interval [a, b], then apply the formula as:-

f(x) = f(a) + f '(a) $\frac{(x-a)}{1!}$ + f '' (a) $\frac{(x-a)^{2}}{2!}$ + f ''' (a) $\frac{(x-a)^{3}}{3!}$ +.....+ f n (ξ) $\frac{(x-a)^{n}}{n!}$.

where ξ = a + θ(x-a) and 0 < θ < 1

Step 2 :

If in the formula we put a = 0, we obtain Maclaurin's formula as:-

f(x) = f(0) + f '(0)(x-a) + f '' (a) $\frac{(x)^{2}}{2!}$ + f ''' (a) $\frac{(x)^{3}}{3!}$ +.....+ f n (ξ) $\frac{(x)^n}{n!}$.

where ξ = θx, 0 < θ < 1

## Examples On Taylor Series Calculator

1. ### Expand the polynomial p(x) = x5 - 2x4 + x3 - x2 +2x -1, in power of the binomial x-1 using the Taylor formula.

Step 1 :

Find the value of the polynomial and its derivetives at the point x = 1.

p(1) = 0, p' (1) = 0

p'' (1) = 0, p'' '(1) = 18,

p'''' (1) = 72, p5 (1) = 120

pn (1) = 0 (n >= 6) at any x

Step 2 :

Substituting the value in the formula:-

p(x) = $\frac{18}{3!}$ (x-1)3 + $\frac{72}{4!}$ (x-1)4 + $\frac{120}{5!}$ (x-1)5

p(x) = 3(x-1)3 + 3(x -1)4 + (x-1)5 .

3(x-1)3 + 3(x -1)4 + (x-1)5

2. ### Applying the Maclaurin formula expand in powers of x (up to x9 inclusive) the function f(x) = ln (1 + x)defined on the interval [0, 1]. estimate the error due to deleting the remainder.

Step 1 :

f(0)  ln 1 = 0, the derivatives of any order of the given function is:--

fn (x) = (-1)n-1 $\frac{(n-1)!}{(1+x)^{n}}$

fn (0) = (-1)n-1 (n-1)!             (n = 1, 2, 3, ....)

Step 2 :

Substituting the derivatives into the Maclaurin formula, we get.

ln (1+x) = x - $\frac{x^{2}}{2}$ + $\frac{x^{3}}{3}$ - ....... + $\frac{x^{9}}{9}$ + R10 (x)

x - $\frac{x^{2}}{2}$ + $\frac{x^{3}}{3}$ - ....... + $\frac{x^{9}}{9}$ + R10 (x)