The discussion revolves around the integration of power series, specifically addressing the treatment of constants of integration and the differences between indefinite and definite integrals. Participants explore the implications of integrating power series and the role of boundary conditions in determining constants.
Participants express differing views on the necessity and treatment of the constant of integration, with no consensus reached on whether it can be assumed to be zero or how to manage constants in definite integrals.
The discussion highlights the dependence on definitions and the context of integration (indefinite vs. definite), which may influence the treatment of constants.
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?