(a) Use the binomial series to find the Maclaurin series of(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f(x)=\sqrt{1+x^2} [/tex]

[tex] f(x)=\sqrt{1+x^2} = \left( 1+x^2 \right) ^{1/2} = \sum _{n=0} ^{\infty} \binom{1/2}{n} x^{2n} = \binom{1/2}{0} + \binom{1/2}{1}x^2 + \binom{1/2}{3}x^6 + \binom{1/2}{4}x^8 + \cdots [/tex]

[tex] f(x)= 1 + \frac{1}{2}x^2 + \frac{\left( \frac{1}{2} \right)\left( -\frac{1}{2} \right)}{2!}x^4 + \frac{\left( \frac{1}{2} \right)\left( -\frac{1}{2} \right) \left( -\frac{3}{2} \right) }{3!}x^6 + \frac{\left( \frac{1}{2} \right)\left( -\frac{1}{2} \right) \left( -\frac{3}{2} \right) \left( -\frac{5}{2} \right) }{4!}x^8 + \cdots [/tex]

[tex] f(x)= 1 + \frac{1}{2}x^2 + \sum _{n=2} ^{\infty} (-1)^{n-1} \frac{1\cdot 3\cdot 5\cdot \cdots \cdot (2n-3)}{n!2^n}x^{2n} [/tex]

(a) Use (a) to evaluate [tex] f^{(10)}(0). [/tex] (answer: 99,225)

I've had some difficulty to see how this result is obtained, since the [tex] x^{2n} [/tex] in the series above seems to give 0 despite the differentiation steps :surprised . Can anybody help me clarify this?

Thanks

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# Homework Help: Confused (differentiation)

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