๐ Maclaurin Series vs. Taylor Series: Key Differences Explained
In calculus, both Maclaurin and Taylor series are powerful tools for approximating functions using polynomials. While they are closely related, understanding their subtle differences is crucial for mastering calculus concepts.
๐ค Definition of Taylor Series
The Taylor series is a representation of a function $f(x)$ as an infinite sum of terms involving its derivatives evaluated at a specific point $a$.
- ๐ The Taylor series expansion of a function $f(x)$ about the point $x=a$ is given by:
- โ๏ธ $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$
- ๐ Where $f^{(n)}(a)$ denotes the nth derivative of $f(x)$ evaluated at $x=a$, and $n!$ is the factorial of $n$.
๐ก Definition of Maclaurin Series
The Maclaurin series is a special case of the Taylor series where the function's derivatives are evaluated at $a=0$.
- ๐ In other words, the Maclaurin series is a Taylor series centered at zero.
- ๐งช The Maclaurin series expansion is given by:
- ๐ $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ...$
๐ Maclaurin Series vs. Taylor Series: A Comparison Table
| Feature |
Taylor Series |
Maclaurin Series |
| Definition |
Representation of $f(x)$ as an infinite sum of terms involving its derivatives evaluated at a point $a$. |
Representation of $f(x)$ as an infinite sum of terms involving its derivatives evaluated at $0$. |
| Center |
Centered at a general point $x = a$. |
Centered at the origin, $x = 0$. |
| Formula |
$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ |
$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$ |
| Relationship |
General form. |
Special case of Taylor series. |
| Application |
Approximating functions around any point. |
Approximating functions around the origin. |
๐ Key Takeaways
- ๐ A Maclaurin series is simply a Taylor series evaluated at $x = 0$.
- ๐ The choice between using a Taylor or Maclaurin series depends on the point around which you want to approximate the function. If you want to approximate around $x = 0$, use Maclaurin. Otherwise, use Taylor.
- ๐ก Both series provide polynomial approximations of functions, which can be very useful in situations where evaluating the original function is difficult or impossible.