Questions about integration of power series

[hyper]

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?

 

[Hootenanny]

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?

Why should it be zero? Upon taking the derivative of S(x) + C, one obtains the original sum s(x).

C is arbitrary by definition, unless it is fixed by a boundary condition.

 

[hyper]

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?

 

[Hootenanny]

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?

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.