Given that sinh(x) = (e^x - e^-x) / 2, find a series representation for arcsinh(x). So my book did a Maclaurin series expansion of sinh(x) = x + x^3 / 3! + x^5 / 5! + ... Then it said: the inverse will have some series expansion which we will write as arcsinh(...
The Taylor's series of finite number of terms with remainder is f(x)=f(a)+f′(a)(x−a)+f″(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n+⋯. The preceding series is known as Maclaurin series for a=0. And the required series is Maclaurin se...
To determine the limit of function by using the Maclaurin series, we need to know the Maclaurin series for some functions. The following Maclaurin series expansion for some of the functions will be useful: {eq}\begin{align} \\ &\hspace{1cm} \sin x = x -\frac{x^3}{3!}...
Maclaurin series: A Taylor series is a series expansion of a function at a point. Consider a functionf(x)about a pointx=ais given by f(x)=f(a)+f′(a)(x−a)+f″(a)(2!)(x−a)2+f‴(a)(3!)(x−a)3+...+fn(a)(n!)(x−a...
Suppose that {eq}f(x) {/eq} has derivatives of all orders at {eq}x=0 {/eq}. The Maclaurin series is then defined to be {eq}\sum_{n=0}^{\infty} \dfrac{f^{(n)}(0)}{n!}x^n {/eq} This is the Taylor series centered at {eq}x=0 {/eq} and is used to approxim...
The Maclaurin Series Expansion:To find the Maclaurin expansion of a function, we need the value of the function itself and its first few higher order derivatives at x=0. Then, we'll use the following formula of the Maclaurin expansion: f(x)=f(0)+xf′(0)1!+x2f″(0)2!...
Find the MacLaurin series for {eq}f(x)=\ln(1-2x). {/eq} On what interval is the expansion valid? Power Series; Radius of Convergence: To find the MacLaurin series for the function {eq}\ln(1-2x), {/eq} we notice that: {eq}\frac{d\left(\ln(1-x...
Power Series: Formula & Examples from Chapter 2/ Lesson 10 30K A power series is an infinite polynomial on the variable x and can be used to define a variety of functions. Explore the formula and examples of power series, discover recommendations and suggestions for using...
{/eq} Then the Maclaurin series of {eq}\displaystyle\int f(x)\; dx {/eq} can be found by integrating the Maclaurin series of {eq}f(x) {/eq} term by term and the radius of convergence of the resulting series is also equal to {eq}R; {/eq} tha...
Calculate the Maclaurin series generated by f\left ( t \right ) = sinh\left ( t \right ). Determine the radius of convergence. Find the Maclaurin series for the following function and determine its radius of convergence R. f(x) =...