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This was a question on a past exam, and I'm very confused about how to do this. We're not allowed calculators, so I'm sure there must be some simple solution that I'm missing.

It asks: Use power series to estimate [tex]\int_{0}^{1/2}\frac{1}{\sqrt{1+x^3}}dx[/tex] within 10^-3.

I started off with the binomial series:

[tex](1+x^3)^{-1/2} = \sum_{k=0}^{\infty}\left(\begin{array}{ccc}

-1/2 \\

k \end{array}\right)x^{3k}[/tex]

I then integrated from 0 to 1/2 to get:

[tex]\int_{0}^{1/2}(1+x^3)^{-1/2}dx = \sum_{k=0}^{\infty}\left(\begin{array}{ccc}

-1/2 \\

k \end{array}\right)\frac{(1/2)^{3k+1}}{3k+1}[/tex]

Now I'm stuck because I don't know how to make it accurate within 0.01. The remainder has a formula:

[tex]R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}[/tex]

But I don't know how to find M, such that [tex]f^{(n+1)}(c)<=M[/tex].

Did I do this one completely wrong? Is there a power series that's easier to work with than the one I came up with? To me, it seems almost impossible to do this question without a lot of tedious work and using a calculator, but our exam is strictly no calculators allowed.

Any help would be appreciated!

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# Estimates using Power Series

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