4

I am so terrible at finding a closed form of a given sequence. Please help me on the following sequence:

$$ f(x) = 1 - \frac{1}{8}x-\frac{7}{128}x^2-\frac{35}{1024}x^3-\frac{805}{32768}x^4-\frac{4991}{262144}x^5-\frac{64883}{4194304}x^6-\frac{435643}{33554432}x^7+.. $$

I realize there are powers of 2 in the denominator but I can not relate the numbers in the whole series.

thanks.

Yakup
  • 41
  • That's not a sequence. That's a function of $x$. – Chris Eagle Mar 02 '13 at 20:48
  • 1
    Not a sequence, a (power) series. – Julien Mar 02 '13 at 20:49
  • Then my question becomes; is there a way to find $$a_{n}$$ in terms of $$n$$? – Yakup Mar 02 '13 at 20:52
  • 1
    @Yakup Given only finitely many terms it is impossible. – Git Gud Mar 02 '13 at 20:55
  • 1
    And how/where did this series appear? – Julien Mar 02 '13 at 20:57
  • I needed to find the critical point of this function by the ratio method. as best as I can –  Mar 02 '13 at 21:08
  • 1
    @Yakup: You can merge your current account with the one you used to post this question originally by doing this: From any page footer -> 'contact us' >> 'Merge user profiles' – Zev Chonoles Mar 02 '13 at 21:09
  • I didn't get a hit in oeis for your numerators but your denominators show up in as A061549. One of the notes is that the sequence is the sequence of demoninators of the power series of $1/(\sqrt{1+x}-\sqrt{1-x})$. Which is presumably the original context for the problem. – JSchlather Mar 02 '13 at 21:23
  • It is difficult without an expression for the general term. True, the denominators are powers of $2,$ but $3,7,10,15,18,22,25$ doesn't have obvious structure. Neither do the numerators $1,7,7\cdot 5, 7\cdot5\cdot3,7\cdot 713?$ – Ross Millikan Mar 02 '13 at 21:24

1 Answers1

5

By inspection, this is the power series for $f(x)=\sqrt[8]{1-x}$ around $x=0$. So, the coefficient of $x^n$ will be $(-1)^n \binom{1/8}{n}$.

David Moews
  • 16,651