Post 4: Deriving Taylor Series using Maclaurin Series

In the previous post, we discussed the Taylor series expansion, which allows us to approximate a function using an infinite sum of terms. However, there is a special case of the Taylor series known as the Maclaurin series, where the expansion point is at 0. In this post, we will explore how to derive Maclaurin series expansions using simplifications when the point of expansion is at 0.

The Maclaurin Series

The Maclaurin series is a type of Taylor series expansion where the expansion point is at 0. It is named after the Scottish mathematician Colin Maclaurin. The general form of the Maclaurin series for a function f(x) is:

Where f'(x), f''(x), f'''(x), and f^(n)(x) represent the first, second, third, and nth derivatives of the function f(x) evaluated at 0, respectively, and n! denotes the factorial of n.

Deriving Maclaurin Series Expansions

To derive the Maclaurin series expansion of a function, we follow similar steps as in the Taylor series expansion. However, since the expansion point is at 0, the simplifications can make the process easier. Let's take a look at an example:

Example 1: Find the Maclaurin series expansion of the function f(x) = e^x.

Solution: We begin by finding the derivatives of f(x):

f'(x) = e^x

f''(x) = e^x

f'''(x) = e^x

f^(n)(x) = e^x, for all values of n

Now, we evaluate these derivatives at 0 to get the corresponding coefficients in the Maclaurin series expansion:

f(0) = e^0 = 1

f'(0) = e^0 = 1

f''(0) = e^0 = 1

f'''(0) = e^0 = 1

f^(n)(0) = e^0 = 1, for all values of n

Plugging these values into the Maclaurin series formula, we have:

f(x) = 1 + (1/1!)(x-0) + (1/2!)(x-0)^2 + (1/3!)(x-0)^3 + ...

Simplifying further, we get the Maclaurin series expansion of f(x) = e^x:

f(x) = 1 + x + (x^2)/2! + (x^3)/3! + ...

This infinite series represents the Maclaurin series expansion of f(x) = e^x.

Conclusion

In this post, we explored the concept of the Maclaurin series, which is a special case of the Taylor series where the expansion point is at 0. We learned how to derive Maclaurin series expansions by evaluating derivatives of the function at 0 and using simplifications. The Maclaurin series provides a way to approximate functions and can be used in various applications in calculus and beyond.