A problem!

Dec 2009
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\(\displaystyle f(x)=x-\frac{3}{2}x^2+\frac{11}{6}x^3-\frac{50}{24}x^4+\frac{274}{120}x^5+...\)
This is Taylor series of function \(\displaystyle f(x)=\frac{ln(1+x)}{1+x}\)

My question is:

\(\displaystyle f(x)=x-\frac{3}{2}x^2+\frac{11}{6}x^3-\frac{50}{24}x^4+\frac{274}{120}x^5+...=\Sigma^\infty_{n=0}(-1)^{n+1}a_n x^n\)

What is form of \(\displaystyle a_n\)?


Thank you very much!
 
Last edited:

Ackbeet

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Well, the denominators look something like a factorial (except for that pesky 3 in the squared term; perhaps you could massage it to look better). Maybe you could set up a recurrence relation to figure out the numerators?
 
Dec 2009
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434
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Ohhh... it's not 2/3 it's 3/2 :(

Sorry!!!

(I'm going to change it in my original post)


Thank you Mr. Keister!
 

Ackbeet

MHF Hall of Honor
Jun 2010
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Aha. So it is a nice little factorial in the denominator. Well, the recurrence relation idea sounds nice, but I'm not sure how it would work in principle. Why don't you show all the details of the computations for those first few terms? Maybe something'll pop out.
 
Dec 2009
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434
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Aha. So it is a nice little factorial in the denominator. Well, the recurrence relation idea sounds nice, but I'm not sure how it would work in principle. Why don't you show all the details of the computations for those first few terms? Maybe something'll pop out.

You ask for computations ?

Here it is: :) Function calculator

Moreover... look here...

1 3 11 50 - OEIS Search Results
 

Ackbeet

MHF Hall of Honor
Jun 2010
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No, that's not what I'm interested in. I'm interested in seeing the step-by-step calculations for computing each coefficient. I'm curious to see if a pattern pops out by examining that process.
 
Dec 2009
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Unfortunately, I can't do so, it is very complicated, but the thing I can do is to post the original question asked.

Q.
Find Taylor polynomial of \(\displaystyle f(x)=\frac{ln(1+x)}{1+x}\) around \(\displaystyle x=0\), find formula for \(\displaystyle f^{(n)}(x)\).
Find the radius of convergence of infinite Taylor polynomial(power series).
 
Dec 2009
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So, the radius of convergence \(\displaystyle R\) of \(\displaystyle \Sigma^{\infty}_{n=0}(-1)^nH_nx^n\) is R=0?


Thanks!
 
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