The Taylor series is a special Power series designed to show mathematicians how power series behave. Now, power functions are special functions with a power greater than 1 and moving towards infinity. The exponential function is a popular power function, but that is not all there is to it. The sin function, cosine function and Hyperbolic functions are different power series.
The Taylor series actually help students know how the different functions mentioned above behave. The Taylor series makes it easy for students to determine the behaviour of the power functions as it represents their infinite sum of terms. For the Taylor series to work, the functions will need to be expressed as derivatives at a sine point.
Using the Taylor series, students will only need to determine the n + 1 terms of the nth polynomial in the power function. For example, students who want to want to determine the derivative of a function polynomial degree of 6, will only need to compute the 6th sum of terms (n + 1) using the Taylor series.
In this piece, students will learn the Taylor Series formula and how to apply it when calculating the approximations of power functions.
What is the Taylor Series?
The Taylor series is an approximation of a function whose accuracy gets better as n increases at a constant point of a.
The Taylor series becomes the Mclaurin series when a = 0.
The Taylor Series Formula
The Taylor Series of Function f(x), which is infinitely differentiable at a centered real or complex number denoted by a is the power series;
If a = 0, the Taylor series becomes the McLaurin Series, another special power series
Both formulas are the same and are sufficient when using the series to determine power function behavior. The first is simply the expanded version of the other. Students will need to expand the Taylor series formula when approximating a function.
How to Calculate the Taylor Series?
To calculate the Taylor series, students will need to know how to differentiate functions using the quotient rule, product rule, & Chain Rule. Students may need to understand the factorial function when solving for the Taylor series as several functions employ it.
- Below are the steps to determining the Taylor series
- Write out the Taylor series formula for easy computation
- Expand the formula up to the fourth series as this is usually considered enough to predict the series limit
- Determine the respective derivatives of f(x) and substitute into the expanded formula
- Simplify as much as possible to arrive at the Taylor Series
What is Needed to Calculate the Taylor Series?
The Taylor Series is generally very easy to calculate as all you need to do is follow the steps of calculation outlined above. However, it is important that you know certain mathematical notations and subjects to successfully get an accurate calculation. Firstly, the summation term simply shows that the series can attain a derivative up to infinity. However, most calculations stop at the fourth derivative as it already signifies the direction of the series.
Secondly, you need to have a very good understanding of differentiation and its different calculation methods. There is the quotient rule which is used for differentiation functions and variables with divisions. The product rule is for calculating the multiplying functions. Thee is also the chain rule which is for computing complex functions.
Thirdly, knowing how to calculate factorials is also very important. However, you will most likely not need to calculate more than the 4! Knowing the calculation will help you understand several divisions when dealing with the Taylor series.
Example On Using Taylor Series
Example 1
Find the Power series using Taylor Series for ex with a centered point of a = 0
Solution
The first step is to write out the formula
This is true because the derivative of an exponential with a coefficient of 1 and a positive x will always be the same.
Substituting the above into the formula will give
Recall that the series is centered at a = o. so substitute o anywhere a is found in the formula
Obviously, any value or variable with a power of 0 will always be equal. So e0 = 1
This Taylor series is known as a McLaurin series because a is centered at 0
Example 2
Solution
Substitute the derived values into the expanded formula
Therefore the Taylor series of
Conclusion
From the above examples, you can see that Calculating the Taylor series is no more than just a sequence of steps and putting the right numbers and variables where they fit. It does not require many complexities on the surface. The only area where you may find it difficult is when you might need to find a complex derivative, which is an entirely different topic.
Generally, when a = 0, the Taylor series is relatively easier to compute as a is naturally eliminated from the expression, making the entire computation straightforward.
