Question

Hey mates, another Series question. Will ask on the next post

Answer 1

\[f(x) =(1-3x)^{-5}\]\[f(x)=\ln{(1-x)}\]Find the MacLaurin representation of both functions and their respective radius of convergence. The first one I am pretty much stuck. The second one I tried to use e^x expansion, but it became way to messy. Any ideas, mates? Thanks in advance

Answer 2

the second one has a neat trick; take the derivaitve and long divide it; then integrate the long dividion to get back to ln(1-x)

Answer 3

If I may ask a personal question you don't have to reply but where are you from, I have not met anyone that says mates besides someone from Australia.

Answer 4

-1/(1-x)


-1-x-x^2-x^3-x^4....
--------------
1-x) -1
(-1+x)
------
-x
(-x+x^2)
---------
-x^2

\[\int -1-x-x^2-x^3-x^4....dx=-x-x^2/2-x^3/3-x^4/4-x^5/5....\]

Answer 5

\[ln(1-x)=\sum_{n=0}^{inf} \frac{-x^{n+1}}{n+1}\]

Answer 6

@ParisWinter Actually, I am from Brazil, so English is not my native language. I tend to use a bit of old of style words because I generally don't like slangs. :-)

Answer 7

Oh that's awesome!!!! I tell people all the time if i could go anywhere i would want to go to Brazil!!!!!

Answer 8

@amistre64 Thank you very much. Gonna try it myself. I am having issues conceptualizing these series expasions, but, well, have to deal with it.

@ParisWinter It's a nice country, but very hot, even in the Southern part of it, haha.

Answer 9

\[\lim_{n\to inf} \frac{-x^{n+1}}{n+1}\frac{n}{-x^{n}}\]
\[\lim_{n\to inf} \frac{nx}{n+1}\]
\[|x|\ \lim_{n\to inf} \frac{n}{n+1}\]
\[|x|*1\]

Answer 10

|x|<1 defines the radius of convergence i believe

Answer 11

or the 1 is the radius :)

Answer 12

True!!! I am from Florida though it gets pretty hot but probably not as hot as Brazil!!!!

Answer 13

@amistre64 Indeed, I got 1 for the radius, but because I cheated and looked up the series expansion. At least, I got part of it right.
Anyway, I can't work it out from e^x expansion? It seemed more natural, yet very messy.

@ParisWinter Ah, Florida. I think that there are a lot of Brazilian in Miami nowadays.

Answer 14

I wouldn't know I haven't been to Miami, I live in Ocala which is in central Florida. I would love to go though very soon!

Answer 15

@ParisWinter Isn't Ocala near Gainesville and McIntosh?

Answer 16

Yes it is!!!

Answer 17

I see. :-) Well, at least it's more or less near Tampa and Daytona. Anyway, gotta study for my exams. See you around, @ParisWinter
And once again, thank you very much @amistre64 Will try to fiddle around the series a bit more.