The Maclaurin series given below for the function f(x)=tan ^(-1)x is tan ^(-1)x=x- (x^3)3+ (x^5)5-⋯ +(-1)^(n+1) (x^(2n-1))(2n-1).If h(x)=f(2x), write the first four non-zero terms and the general term of the Maclaurin series for h(x).相关知识点: 试...
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Maclaurin series and general calculus question Homework Statement This question has four parts which may follow up from each other so I incuded all the parts. The real problem I'm having is with d Consider the function f ang g given by f (x)=( e^x+[e^-x])/2 & g (x) =( [e...
The general Maclaurin series formula in summation form looks like this: f(x)=∑n=0∞fnan!(x−a)n. The n stands for the number of the term (the n value of the first term is 0). The n! is the factorial of n, or the product of n and all the numbers less than n to 1, ...
The Taylor series of cos(x) can be found by taking derivatives of cosine and plugging them into the general equation for a Taylor series, using the fact that the derivatives repeat themselves in groups of four. Does the Maclaurin series for cos x converge for all x? Yes, the Maclaurin se...
The problem of interpolation led Euler to formulate the problem of integration, i.e., to express the general term of a series by means of an integral. The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was ...
Similarly let’s add a biquadratic term (degree 4). Therefore, our computations will look like, If you look closely, the series follows a pattern, and it can be written as So, the general form for the Maclaurin series is, Figure 2 General form for Maclaurin Series General form for Ta...
TANGENT functionBERNOULLI numbersDIFFERENTIABLE functionsBESSEL functionsZETA functionsIn view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expressi...
Although this series is clearly convergent for allx, as may be verified using the ratio test, it is instructive to check the remainder term,Rn. From Eq.(2.29)we get Rn(x)=xn+1(n+1)!f(n+1)(ξ)=xn+1(n+1)!eξ, whereξis between0andx. Irrespective of the sign ofx, ...
For the given product of the different functions, we'll use the general Maclaurin series of sine trigonometric function and exponential function. {eq}e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots\\ \sin x= x-\frac{x^3}{3!...