Question

Need some help understanding properties of functions defined by power series

Answer 1

Do you have a specific question?

Answer 2

The general form they give for integeration of a power series is:\[\int\limits f(x)dx=C+\sum_{n=0}^{\infty}a _{n}\frac{ (x-c)^{n+1} }{ n+1 }=c+a_{0}(x-c)+a _{1}(x-c)^{2}+....\]
how do I apply something like that to a generic nth term?

Answer 3

ex f(x)=\[\sum_{n=1}^{\infty}\frac{ x ^{n} }{ n }\]

Answer 4

how would I find the anti-derivative of this function?

Answer 5

would it be just generically......\[\int\limits f(x)dx=\sum_{n=1}^{\infty}\frac{ x ^{n+1} }{ n(n+1) }\]

Answer 6

@amistre64 could you help

Answer 7

yes

Answer 8

the part I don't get is where in their general form they have the a sub nth term of the sequence times the anti-derivative.

Answer 9

since the only x parts in the power series come for the exponentiated "x" the rest is just constants

Answer 10

sorry it shoudl have said that I don't understand why they have the a sub nth term of the SERIES times the anti derivative

Answer 11

\[f(x)=\sum_k c_nx^n\]
\[\int f(x)dx=\int\sum_k c_nx^n~dx\]
\[F(x)+c=\sum_k \frac{c_n}{n+1}x^{n+1}\]

Answer 12

all a series is is a polynomial of so many terms; integrating a sum is equal to integrating each term

Answer 13

So then do I need to multiply the anti-derivative times the original a sub nth term?

Answer 14

of course there are certain convergence issues to address, but overall ..

Answer 15

a_n is a constant ...

Answer 16

ok so then it would just be using properties of a series where you can multiply a constant to a series after the series is evaluated?

Answer 17

Then the original radius of convergence still applies but I would have to evaluate the end points to determine if they are in the interval of convergence?

Answer 18

I would have to evaluate them in the antiderivative series to see if they converge or diverge right?

Answer 19

\[f(x)=\sum c_nx^n=c_0+c_1x+c_2x^2+...+c_nx^n+...\]
\[\int f(x)=\int(c_0+c_1x+c_2x^2+...+c_nx^n+...)\]
\[\int f(x)=\int c_0+\int c_1x+\int c_2x^2+\int ...+\int c_nx^n+\int ...\]

Answer 20

im not read up on how convergences effect it, but for the most part, you can just assume it until yelled at later :)

Answer 21

lol thanks

Answer 22

yw