Taylor Series Approximation: Definition, Formula, and Example

Taylor Series Approximation: Definition, Formula, and Example

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Aug 29, 2023 14:08 IST

Let f(x) be a continuous and infinitely differentiable function (i.e., a function that can be differentiated infinite times), then the taylor series of f(x) is a series expansion of f(x) about a point x = a. In this article, we will discuss how to find the approximation of a function using Taylor series.

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Taylor series is a series expansion of a function around a point that can be utilized to approximate a function. It is an application of derivatives of a function that has a vast application in machine learning and deep learning.
In machine learning, to calculate the gradient descent, we try to minimize the cost function using differentiation. However, not all the functions can be differentiated, so here we use the Taylor series to approximate the function.
Taylor series expansion is applied in the deep neural network to optimize the performance measure is one of the most important uses of the Taylor series in deep learning.
In Neural Networks, the Taylor series can be applied to isolate the difficulties like shattered gradients. It can also be used to explain the neural network’s prediction quantitatively.
In this article, we will briefly discuss the Taylor series, its formula, its proof, and some of the common Taylor series expansions.

Table of Content

Before starting the taylor series, let’s discuss one important series that we will use to prove the taylor series formula, i.e., Power Series.

Power Series: Power series about the center x = a is an infinite sum of the form

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where

ci: constant coefficients

What is the Taylor Series?

Let f(x) be a continuous and infinitely differentiable function (i.e., a function that can be differentiated infinite times), then the taylor series of f(x) is a series expansion of f(x) about a point x = a.

The Taylor series expansion of any function f(x) at x = a is given by:

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This can also be written as:

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Note: If in the above formula, a = 0, then the above expansion is known as Maclaurin Series.

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Proof of Taylor’s Theorem

Let f(x) be a polynomial function, such that:

f(x) = a0 + a1x + a2x2 + a3 x3 + a4x4 +….. + ak xk+……,         ……. (1)

Now substituting x = 0, in (1) we get:

f(0) = a0

=> a0 = f(0)/0! …..(i)

Differentiating f(x), with respect to y, we get:

f’(x) = a1 + 2a2x + 3a3x2 + 4a4x3 + ….,    …….(2)

Substituting x = 0, in (2) we get:

f’(0) = a1

=> a1 = f’(0)/1! …..(ii)

Differnetitaing f’(x), with respect to x we get:

f’’(x) = 2a2 + 6 a3 x + 12a4 x2 + ….,      …..(3)

Substituting x = 0 in equation (3), we get:

f’’(0) = 2a2

=> a2 = f’’(0)/ 2

=> a2 = f’’(0)/ 2!….(iii)

Differnetitaing f’(y), with respect to y we get:

f’’’(x) = 6a3 + 24a4 x + ….., …………(4)

Substituting x = 0 in equation (4), we get:

f’’’(0) = 6a3

=> a_3 = f’’’(0)/ 6 

=> a3 = f’’’(0)/ 3!…..(iv)

Generalizing, the result of (i), (ii), (iii), and (iv) we get:

an = fn(0) /n!……(v)
Now, substituting the value of (i), (ii), (iii), (iv), and (v) in (1), we get:

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Now, generalizing the above we get:

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Example

Example -1: Find the Taylor series of sin x for x = 0.

Answer: Here, f(x) = sin x, and we know sin x is an infinitely differentiable function that can be represented as:

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Now, differentiating f(x), we get

f(x) = sin x => f(0) = 0

f’(x) = cos x => f’(0) = 1

f’’(x) = -sin x => f’’(0) = 0

f’’’(x) = -cos x => f’’’(0) = -1

f(iv) (x) = sin x => f^(iv)(0) = 0

=> f(iv)(x) = f(x)

So, continuing the differentiation we will get the repetition only

Now, substituting the above values in the Taylor series formula, we get:

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Example – 2: Find the Taylor series expansion of f(x) = 1/x at x = 1.

Answer: here, f(x) = 1 / x, and we know 1/ x is an infinitely differentiable function except x = 0 and the graph of f(x) = 1/x looks like: 

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Now, differentiating f(x), we get

f(x) = 1/x => f(1) = 1/1

f’(x) = -1/x^2 => f’(1) = -1/1=-1

f’’(x) = 2/x^3 => f’’(1) = 2/1^3=2

f’’’(x) = -6/x^4 => f’’’(1) = -6/1^4=-6

Now, substituting the above values in the Taylor series formula, we get:

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Now, we will discuss some commonly use Taylor Series Expansions:

Commonly used Taylor Series Expansion

Exponential Function

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Logarithmic Function

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Trigonometric Function

sin(x)

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cos(x)

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tan (x)

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Conclusion

In this article, we have briefly discussed Taylor series approximation, steps to find the approximation of a function and application how Taylor series can be used in machine learning and deep learning.

Hope you will like the article.

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About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio