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I want to know what is the Taylor series of $\sqrt{1+x^2}$ while $x \to 0$.

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closed as off-topic by 900 sit-ups a day, Miha Habič, Jyrki Lahtonen, Adam Hughes, Sami Ben Romdhane Jul 30 at 8:00

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This looks like homework. Please consider tagging it as homework. –  Aryabhata Oct 23 '10 at 16:56
@moron no it's not homework I want it to prove something –  Am1rr3zA Oct 23 '10 at 17:16
en.wikipedia.org/wiki/… –  Qiaochu Yuan Oct 23 '10 at 17:34
I really don't like this one: this site is not a calculator! Why can't you compute it yourself? As this is stated, this is not even a question... I want coffee! –  Mariano Suárez-Alvarez Oct 23 '10 at 21:28

2 Answers 2

up vote 4 down vote accepted

If you just want the answer, you can find it at WolframAlpha

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link is broken –  anonymous Oct 23 '10 at 17:24
I tried putting the link in using the link command and it didn't take. So I put the full link in, but it breaks at the ^. If I copy the whole thing and paste it into the browser, it works. Best I seem to be able to do. I have posted links in successfully before. –  Ross Millikan Oct 23 '10 at 17:28
You need to use %5e for ^. See meta.stackexchange.com/questions/60211/…. –  KennyTM Oct 23 '10 at 17:34
@KennyTM Fixed now. Thanks. –  Ross Millikan Oct 23 '10 at 17:40

If you want to work a little bit, you could learn about the binomial series:

$$ (1+x)^\alpha = \sum_{n=0}^\infty \begin{pmatrix}\alpha \\ n\end{pmatrix} x^n $$

and then:

  1. Take $\alpha = \frac{1}{2}$, and
  2. Replace $x$ by $x^2$.

You'll get:

$$ \sqrt{1 + x^2} = \sum_{n=0}^\infty \begin{pmatrix} 1/2 \\ n\end{pmatrix} x^{2n} $$

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