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# Taylor Series Calculator

Enter a function:

W.R.T ?

Enter a point:

Order n:

Calculate the series and determine the error at that point (optional):

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The Taylor series calculator calculates all coefficients of a Taylor series expansion of a function centered at point “n”. Also, you can set point “n” as zero (0) to get the Maclaurin series representation.

## What is the Taylor Series?

In mathematics, the Taylor series is defined as the representation of a given function. It is an infinite series that represents the value of the derivative of a function at a certain point.

There is a special case of the Taylor and Maclaurin series calculator. This series helps to reduce the number of mathematics proofs and is used for power flow analysis.

## Taylor Series Expansion Formula:

The formula used by the Taylor series formula calculator for calculating a series for a function is given as:

$$F (x) = âˆ‘^ âˆž_{n=0} f^k (a) / k! (x â€“ a) ^k$$

Where

f^(n)(a) = nth order derivative of function f(x), as evaluated at x = a

n =Â  Â where the series is centered.

The series will be most precise near the centering point.

A Taylor expansion may be infinite, but we can select to make our series or function as little or long terms as we want. We can set the maximum n value to make it an n-order series.

## Example:

Calculate Taylor expansion of (x^2+4)^{1/2} up to n = 4?

### Solution:

#### Maclaurin Equation for the Function:

Given function f(m)= (x^2+4)^{1/2} and order point n = 1 to 4

Maclaurin equation for the function is:

$$f(y)=âˆ‘k=0^âˆž f (k) (a)/ k! (x â€“ a)^k$$

$$f(y)â‰ˆ P (x) = âˆ‘_k=0^4 f^(k) (a) / k! (x â€“ a)^k = âˆ‘k=0^âˆž f (k) (a)/ k! (x â€“ a)^k$$

$$F^0(y) = f (y) = \sqrt {x^2 + 4 }$$

#### Take the First Derivative:

$$f (1) = \sqrt {5}$$

Take the first derivative $$f^1(y) = [f^0(y)]’$$

$$[\sqrt {x^2 + 4 }]’ = \frac {x} { \sqrt {x^2 + 4 }}$$

$$(f (1))’ = \frac { \sqrt { 5}} {5}$$

#### Take the Second Derivative:

$$f^2 (y) = [f^1 (y)]’ = \frac {x} { \sqrt {x^2 + 4 }} = 4 / (x^2 + 4) ^{3/2}$$

Calculate the second derivative at a given point:

F (1)â€™â€™ = 4\sqrt{5} / 25

#### Take the Third Derivative:

$$f^3(y) = [f^2(y)]’ = (4/ (x^2 + 4) ^{3/2}) = – 12x / (x^2 + 4) ^{5/2}$$

Calculate the third derivative of $$(f (0))”’ = – 12 \sqrt {5} / 125$$

#### Take the Fourth Derivative

$$f^4 (y) = [f^3 (y)]’ = [- 12x / (x^2 + 4) ^{5/2}]’ = 48x^2 â€“ 48 / \sqrt {x^2 + 4} (x^6 + 12x^4 + 48x^2 + 64)$$

Then, find the fourth derivative of the function (f(0))”” Â = 0

$$f(y) â‰ˆ \sqrt {5}/0! (x â€“ 1)^0 + \frac { \sqrt{5} / 5} {1!} (x â€“ 1)^1 + â€¦ + 0 / 4! (x â€“ 1)^4$$

#### Final Result:

After simplification, we get the final results:

$$f(y) â‰ˆ P(x) = \sqrt {5} + \sqrt {5} (x-1) / 5 + 2 \sqrt {5} (x-1)^2 / 25 â€“ 2 \sqrt {5} (x – 1)^3 / 125Â$$

All the steps are specifically solved and represented in the Taylor series calculator calculations to make them easy to understand.

## How Does Calculator Work?

### Input:

• Firstly, substitute a function with respect to a specific variable.
• Now, enter a particular point to evaluate the Taylor series of functions around this point.
• Then, add the order n for approximation.
• By using this Taylor series error calculator, find the series and determine the error at the given point. (optional)
• Click the calculate button for further solutions.

### Output:

• The sum of the Taylor series calculator with steps shows the series after simplification.
• It computes the series of entered functions around the given order number n.
• The third-degree Taylor polynomial calculator takes the derivative for getting the polynomials and puts the results into the Taylor series formula.
• It displays the results after the simplification of polynomials.

## Reference:

From the source of Wikipedia: Analytic functions, Approximation error, convergence, Generalization, List of Maclaurin series of some common functions, Exponential function, Natural logarithm, Geometric series, Binomial series.