Question

What is the coefficient of x^3 in the taylor series (around x=0) of: ln(x+1)/ln(x-1)

I know that ln(x+1) ~ x-(x^2)/2-(x^3)/3
any way to do this without headaches?

Answer 1

without headaches? prolly not. sadistic math teachers are just mean

Answer 2

product rule tends to be a nicer fit than quotient rule, but still takes time

ln(a) [ln(b)]^-1

Answer 3

this question is from a test :( [multi choices].. I guess there is an easier way instead of deriving this 3 times..

Answer 4

0: ln(a) [ln(b)]^-1
1: a'/a [ln(b)]^-1 - ln(a) [ln(b)]^-2 b'/b
2: ....

maybe, but 3 times deriving seems to take a shorter time to me than waiting for someone else to do it :)

Answer 5

1: a' a^-1 [ln(b)]^-1 - ln(a) [ln(b)]^-2 b'b b^-1
2: a'' a^-1 [ln(b)]^-1
- a'a' a^-2 [ln(b)]^-1
- a' a^-1 [ln(b)]^-2 b'/b
- a'/a [ln(b)]^-2 b' b b^-1
- ln(a) [ln(b)]^-2 b'' b b^-1
- ln(a) [ln(b)]^-2 b'b' b^-1
+ ln(a) [ln(b)]^-2 b'b' b b^-2
+2 ln(a) [ln(b)]^-3 b' b b^-1 b' b^-1

that might reduce lol ... but yeah, prolyy an easier way

Answer 6

if we know the taylors for top and bottom, we could just divide them out right?

Answer 7

that's what i'm thinking of.. the top is a known taylor expansion. the bottom is minus the first? I have no idea about the ln(x-1)..

Answer 8

not sure what michael is doing, maybe using your post to practice latex?

Answer 9

i have reported him >.< bah

Answer 10

wolfram says the coefficient is 5/12 ..

I hate math teachers. They are mean :'(

Answer 11

ln(x-1) derives to:

1/(x-1) ..

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

integrated to: -x -x^2/2 -x^3/3 ..... would be the taylor for the bottom i believe

Answer 12

x -x^2/2 +x^3/3 -x^4/4 ...
------------------------
-x -x^2/2 -x^3/3 -x^4/4...

i first notice that we can factor out an x ....

1 -x/2 +x^2/3 -x^3/4 ...
------------------------
-1 -x/2 -x^2/3 -x^3/4...


then work some longhand till you get to the x^3 ?

Answer 13

-1 +x -
-------------------------
-1 -x/2 -x^2/3 -x^3/4... | 1 -x/2 +x^2/3 -x^3/4 ...
-1 -x/2 -x^2/3 -x^3/4...
-------------------------
-x +0 -x^3/2
x +x^2/2 +x^3/3 +x^4/4
--------------------------
x^2/2 ....

Answer 14

http://www.wolframalpha.com/input/?i=%281+-x%2F2+%2Bx%5E2%2F3+-x%5E3%2F4%29%2F%28-1+-x%2F2+-x%5E2%2F3+-x%5E3%2F4%29

well, that does work out .... notice on the series expansion we get the 5/12

Answer 15

is it easier? dunno, maybe on paper

Answer 16

Ahh.. stupid question. Thanks a lot mate! :)

Answer 17

good luck :)