Why should it be zero? Upon taking the derivative of S(x) + C, one obtains the original sum s(x).hyper said:Lets say we have this series:
a0+ a1(x-k)^1 +a2(x-k)^2 +a3(x-k)^3 = s(x)
If I integrate the series a theorem in the books says that I will get the antiderivate S(x)+C, but won't C allways be equal to zero?
Note that in this case you are integrating between limits, i.e. you are evaluating the definite integral, whereas in the previous case you were evaluating the indefinite integral.hyper said:I read this in another theorem:
f(x)= (sigma from n=0 to eternity) an(x-b)^n an is like a0 a1 a2 a3 etc
then:
integrate from b to x f(t) dt= (sigma from n=0 to eternity) an/(n+1)* (x-b)^(n+1)
Here it is no constant, how can I keep track of the constants?
A power series is an infinite series in the form of ∑n=0∞ cn(x-a)n, where cn are the coefficients, x is the variable, and a is the center of the series. It is used to represent a function as a sum of polynomial terms.
To integrate a power series, you can use the standard integration techniques such as the power rule, substitution, and integration by parts. First, you need to find the interval of convergence of the series. Then, you can integrate each term of the series using the appropriate integration method. Finally, you can sum up the integrated terms to get the integrated power series.
The interval of convergence of a power series is the set of values of x for which the series converges. It can be determined using the ratio test or the root test. The interval may include the center of the series or may be centered around the center point.
You can use the Taylor series, which is a type of power series, to approximate a function. The Taylor series is centered around a point and gives an infinite polynomial expression for the function. By truncating the series at a certain term, you can get a polynomial approximation of the function that becomes more accurate as you include more terms.
Power series have many practical applications in fields such as physics, engineering, and economics. They can be used to model various phenomena, such as the motion of objects, electrical circuits, and economic growth. They are also used in numerical analysis to solve differential equations and in computer graphics to create smooth curves and surfaces. Additionally, they are used in calculus to evaluate integrals of non-polynomial functions.