Ln(2+x)

: Math 2730 Sequences and Series: Ln(2+x)
   By Candice Rivet on Monday, March 24, 2008 - 09:53 pm: Edit Post

For ln(2+x) can this be written as 2ln(1+x) or is it ln2 + ln(1+x)?


   By Sasho on Monday, March 24, 2008 - 10:21 pm: Edit Post

Neither! Your second attempt is closer to the target. Start with ln(2+x)=ln[2(1+x/2)].


   By Ka Hei Stephanie Ng on Tuesday, March 25, 2008 - 12:26 am: Edit Post

Can i write ln(2+x) as ln[1+(1+x)] ?? would this work the same as ln[2(1+x/2)] ??


   By Sasho on Tuesday, March 25, 2008 - 07:53 am: Edit Post

You can certainly write ln(2+x) as ln[1+(1+x)], but then, what will you do with that? That would be only useful to get Taylor representation around a=-1, not so for Maclaurin representation.


   By Candice Rivet on Tuesday, March 25, 2008 - 04:29 pm: Edit Post

Ok but if I have ln(2+x)=ln[2(1+x/2)] does this equal ln2 + ln(1+x/2)?


   By Sasho on Tuesday, March 25, 2008 - 04:33 pm: Edit Post

It does.


   By Candice Rivet on Tuesday, March 25, 2008 - 04:50 pm: Edit Post

So then for ln(2+x)*Tan-(x^2) I get [ln2 + (x + x^2/2 +x^3/3 ...)]*(x^2 + x^6/3 + x^10/5...)

DO I multiply ln2 with (x^2 +...) and end with

x + (x^2/2 + ln2x^2) + x^3/3

or am I missing something?


   By Sasho on Tuesday, March 25, 2008 - 04:58 pm: Edit Post

You multiply series like polynomials: each term with each term. So, you should not end up with you wrote above.


   By Jocelyne Chartier on Tuesday, March 25, 2008 - 05:20 pm: Edit Post

Candice, I would check your numbers for expanding ln(2+x)*tan-(x^2) because they are off. ln (2+x) is alternating too. :-)


   By Candice Rivet on Tuesday, March 25, 2008 - 05:31 pm: Edit Post

In my notes I have ln(1+x)=sum of x^n/n
is this not right?


   By Jocelyne Chartier on Tuesday, March 25, 2008 - 05:40 pm: Edit Post

ln(1+x)= Sigma (-1)^(n+1) x^n/n (sigma is from 1 to infinity)

also tan-1(x) = (-1)^n x^(2n+1)/ (2n+1)

so they both will be alternating in sign.

I had the same numbers as you for tan-(x^2)but with alternating signs

but for ln(2+x) i had (ln 2 + x/2 - x^2/8 + x^3/24 - ...)


   By Candice Rivet on Tuesday, March 25, 2008 - 05:49 pm: Edit Post

okay thanks a lot, that helps.


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