 # Taylor Expansion of e^x – Definition and Examples 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$+ … Figure-1. ## Properties ### The radius of Convergence The radius of convergence for the Taylor series of$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$. The error between the actual function and its approximation is bounded by the remainder term. For the Taylor series of$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$for fast computation and approximation of other functions. Exponential functions and their derivatives appear in various machine learning algorithms, especially in loss functions and activation functions in neural networks. ### Chemistry The exponential function, and by extension its Taylor series, plays a role in modeling the kinetics of certain chemical reactions, especially first-order reactions. ## Exercise ### Example 1 #### Approximation Using the First Few Terms Approximate$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.

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