The series e^a ≈1+a+(a^2)/(2 !)+(a^3)/(3 !)+⋯+(a^n)/(n !) where n !=1 ·2 ·3 ·4 ·⋯·n, can be used

step 1 In this question, it is given, E -raged 2 -a is equal to 1 plus A plus A -square -by -factorial

The series e^a ≈1+a+(a^2)/(2 !)+(a^3)/(3 !)+⋯+(a^n)/(n !)...

The series $$ e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\cdots+\frac{a^{n}}{n !} $$ where $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n,$ can be used to approximate the value of $e^{a}$ for any real number $a$. Use the first eight terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator. (a) $e$ (b) $e^{-1}$

The series
$$
e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\cdots+\frac{a^{n}}{n !}
$$
where $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n,$ can be used to approximate the value of $e^{a}$ for any real number $a$. Use the first eight terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator.
(a) $e$
(b) $e^{-1}$

Key Concepts

Truncation Error

When using an infinite series like the Taylor or Maclaurin series to approximate a function, truncation error is the difference between the true value of the function and the sum of the finite number of terms used in the approximation. Understanding truncation error helps in assessing the accuracy of the approximation and deciding how many terms are necessary for a desired precision.

Exponential Function

The exponential function, e^x, is a fundamental and frequently encountered function in mathematics, known for its unique property of being its own derivative. The series expansion of the exponential function is especially useful for numerical approximations and theoretical analysis in fields ranging from calculus to differential equations.

Factorial Function

The factorial function, denoted as n!, represents the product of all positive integers from 1 to n. In the context of series expansions such as that of the exponential function, factorial terms in the denominator play a critical role in ensuring that the contribution of higher-order terms diminishes, allowing the series to converge.

Taylor Series Expansion

The Taylor series allows a function to be expressed as an infinite sum of terms calculated from the values of its derivatives at a single point. For functions like e^x, this series provides a method to approximate the function by adding terms of increasing order, which is essential in both theoretical analysis and practical numerical computation.

Maclaurin Series

A Maclaurin series is a special form of the Taylor series that is centered at zero. It is particularly useful for functions such as the exponential function, where the series takes the form 1 + x + x^2/2! + x^3/3! + ... . This form simplifies calculations and is widely used for approximating the values of functions with inputs close to zero.

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