chromium1387
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Homework Statement
a. Find a power series expansion for arcsin(x) centered at 0.
b. Find the radius of convergence and interval of convergence of the power series in a.
c. Choose an appropriate value of x to plug into the power series found in a. to find a series that converges to \frac{\pi}{2}.
Homework Equations
binomial theorem
ratio test
The Attempt at a Solution
a. Using the Binomial Theorem, I found a power series representation for \frac{1}{\sqrt{1-x^2}} and integrated that to find a power series for arcsin. What I got was: x+\sum\frac{1*3*5*...*(2n-1)x^{2n+1}}{(2n+1)(2^{n})(n!)}
I'm fairly sure this is correct.
b. However, when I go and use the ratio test, I am a bit confused. I get it simplified down to :
|\frac{x^{2}n}{(n+1)(2n-1)}| \rightarrow 0 as n \rightarrow \infty
But what about this x in front of my sum?
If I just left it here, R=\infty and the interval of convergence would be (-\infty,\infty), correct?
c. And I have no idea how to do this one...
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