The **Taylor expansion of e^x** is a cornerstone of calculus, allowing us to express functions as infinite series. Join us on this mathematical journey to understand the infinite power of **$e^x$** through its elegant series representation.

## Definition of Taylor Expansion of e^x

The **Taylor expansion of $e^x$** centered at 0 is defined by:

$e^x$ = 1 + x + $(x^2/2!)$ + $(x^3/2!)$ + $(x^4/2!)$ + …

In general:

$e^x$ = $∑^{∞}_{n=0} (x^n / n!)$

This is also known as its Maclaurin series.

Generally, the **Taylor expansion** of a function** f(x)** about a point a is defined by:

f(x) = f(a) + f'(a)(x – a) + (f”(a)/2!) $(x – a)^2 + (f”'(a)/3!) $(x – a)^3$ + …

Figure-1.

## Properties

**The radius of Convergence**

The r**adius of convergence** for the **Taylor series of $e^x$** is infinite. This means the series converges for all real values of x.

**Error Bounds**

If we truncate the **Taylor series** after a finite number of terms, we’ll have an approximation to **$e^x$.** The error between the actual function and its a**pproximation** is bounded by the remainder term.

For the **Taylor series of $e^x$,** the Lagrange form of the remainder is given by: **$R_n(x)$ = $(e^c / (n+1)!) x^{(n+1)}$** where c is some number between **0** and **x**. Since** $e^c$** is the largest value **$e^x$** can take, the error is bounded by the magnitude of the next term in the series.

**Super-exponential Growth**

The **Taylor expansion of $e^x$** gives insight into the rapid (or super-exponential) growth of this function. For large values of** x,** the terms in the expansion become significantly large.

**Functional Properties**

The **Taylor series of $e^x$** reflects many properties of the exponential function itself. For instance, the property **$e^{(x+y)}$ = $e^x$ * $e^y$** can be seen by multiplying two **Taylor series of $e^x$ and $e^y$**.

**Applications in Differential Equations**

The **Taylor series of $e^x$** is instrumental in solving differential equations, especially linear differential equations with constant coefficients.

It helps transform** differential equations** into algebraic ones in the context of generating functions.

**Complex Extension**

The **Taylor expansion of $e^x$** can be extended to complex numbers. When **x = iθ** (where i is the imaginary unit and θ is a real number), Euler’s formula is derived: $**e^{(iθ)}$ = cos(θ) + isin(θ).** This relationship bridges the exponential function with trigonometric functions through the **Taylor series**.

**Applications **

**Mathematics**

The **Taylor series** helps in solving **differential equations**, especially in the context of power series solutions for ordinary differential equations.

**Quantum Mechanics**

The **exponential function**, the Taylor expansion of **$e^{(ix)}$** plays a role, in comprehending wavefunctions **quantum states** and the **Schrödinger equation**.

**Thermodynamics**

In **mechanics and thermodynamics,** it is common to use the **Taylor series approximation** to calculate partition functions and explore gas properties.

**Engineering**

The **Laplace transform,** which employs** $e^{(st)}$** (a variant of $e^x$), is widely used in **control systems engineering** to analyze and design control systems. The **Taylor series** provides an understanding of how systems behave when they’re close, to a particular **operating point.** When analyzing **circuits** that involve capacitors and inductors it is common to rely on the function and its series.

**Economics and Finance**

Continuously **compounded interest** in finance is based on the **exponential function**. Exponential functions and their **Taylor expansions** are sometimes used in modeling long-term economic growth or decay.

**Biology**

The **growth of populations**, especially under unlimited resources, is often modeled using **exponential growth**. The **series expansion** aids in understanding slight variations and **perturbations around steady states**.

**Computer Science**

Some **algorithms,** especially those used in **numerical methods,** leverage the **Taylor expansion of $e^x$** for fast computation and **approximation** of other functions.** Exponential functions** and their **derivatives** appear in various **machine learning algorithms**, especially in loss functions and activation functions in neural networks.

**Chemistry**

The **exponential function**, and by extension its Taylor series, plays a role in **modeling the kinetics** of certain c**hemical reactions**, especially first-order reactions.

**Exercise **

**Example 1**

**Approximation Using the First Few Terms**

Approximate **$e^{(0.5)}$** using the first four terms of the **Taylor expansion of $e^x$**.

Figure-2.

**Solution**

The **Taylor series of $e^x$** around 0 is:

$e^x$ = 1 + x + $(x^2)$/2! + $(x^3)$/3! + …

For x = 0.5, using the first four terms:

$e^(0.5)$ ≈ 1 + 0.5 + $(0.5^2)$/2! + $(0.5^3)$/3!

$e^(0.5)$ ≈ 1 + 0.5 + 0.125 + 0.0208333

$e^(0.5)$ ≈ 1.6458333

**Example 2**

**Error Estimation**

Using the **Taylor expansion of $e^x$**, estimate the maximum error when approximating **$e^{(0.2)}$** using the first three terms of the series.

Figure-3.

**Solution**

The** Taylor series of $e^x$** is:

$e^x$ = 1 + x + $(x^2)$/2! + $(x^3)$/3! + …

The error $(e^c)/(n+1)! x^{(n+1)}$ can be estimated using the (n+1)-th term:

$(e^c)$/(n+1)! $x^{(n+1)}$

Where c is between 0 and x.

For x = 0.2 and n = 2:

$(e^c)$/(3!) $R_2$(0.2) = $(e^c)$/(3!) $e^c (0.2^3)/3$

Since $e^c$ is maximum when c = 0.2, the error is maximized at:

$(e^c)/(3!) R_2(0.2) = (e^(0.2))/(3!) e^(0.2) (0.2^3)/3$

This is the maximum error for the approximation using the first three terms.

*All images were created with GeoGebra.*