Series expansion for ex

Understanding ex

The symbol “e” represents a mathematical constant approximately equal to 2.71828.
“ex” refers to the exponential function with base e.
Within calculus and analytic geometry, this function is of great importance due to its unique properties.
The derivative of ex is also ex, which is the only function with this property.

The Maclaurin series is used to approximate functions around the point 0.
This series has applications across many areas of mathematics, including calculus and differential equations.
The function ex can be expressed as a Maclaurin series.

The Maclaurin series expansion for ex is given by: 1 + x + x²/2! + x³/3! + x⁴/4! + ….
This series is actually infinite; it has infinitely many terms.
Despite being infinite, it converges for all values of x. This means the series has a definite sum for any value of x.
The general term of the series is given by xⁿ/n!, where n is the term number starting from 0.

Factorial notation (n!) is a fundamental concept for dealing with the series expansion for ex.
The series expansion can be used for approximating the value of ex for small x.
For larger values of x, the series still holds, but more terms would be needed for a precision approximation.

When calculating the value of ex using the series, the more terms you include, the more accurate the approximation.
While performing algebraic calculations, remember the series form of ex. It can often simplify problem-solving.
The series expansion for ex can be used as a foundation for understanding expansions of more complex functions like sin x and cos x.
Practice using the series expansion not only for calculating, but also for understanding the analytic solutions of equations and differential equations.