# How do you find a power series converging to #f(x)=arcsin(x^3)# and determine the radius of convergence?

with radius of convergence

Start from the binomial series:

simplifying:

Applying the ratio test we see that:

Note that:

so we can write the series also as:

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To find a power series representation for ( f(x) = \arcsin(x^3) ), we can use the known power series representation for ( \arcsin(x) ):

[ \arcsin(x) = \sum_{n=0}^{\infty} \frac{1}{2^n}\binom{2n}{n} \frac{x^{2n+1}}{2n+1} ]

First, we replace ( x ) with ( x^3 ) to get the power series for ( \arcsin(x^3) ):

[ \arcsin(x^3) = \sum_{n=0}^{\infty} \frac{1}{2^n}\binom{2n}{n} \frac{(x^3)^{2n+1}}{2n+1} ]

[ = \sum_{n=0}^{\infty} \frac{1}{2^n}\binom{2n}{n} \frac{x^{6n+3}}{2n+1} ]

Now we have the power series representation for ( f(x) = \arcsin(x^3) ):

[ f(x) = \sum_{n=0}^{\infty} \frac{1}{2^n}\binom{2n}{n} \frac{x^{6n+3}}{2n+1} ]

The radius of convergence ( R ) for this power series can be found using the ratio test:

[ R = \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}} \right| ]

where ( a_n ) is the coefficient of ( x^n ) in the power series.

In this case, ( a_n = \frac{1}{2^n}\binom{2n}{n} \frac{1}{2n+1} ) for ( n = 6n+3 ). So,

[ R = \lim_{n \to \infty} \left| \frac{1/2^n \binom{2n}{n} \frac{1}{2n+1}}{1/2^{n+1} \binom{2(n+1)}{n+1} \frac{1}{2(n+1)+1}} \right| ]

Simplifying the expression inside the limit, we get:

[ R = \lim_{n \to \infty} \left| \frac{\binom{2n}{n}}{\binom{2(n+1)}{n+1}} \frac{2(n+1)+1}{2n+1} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{(2n)!(n+1)!(2n+2)!}{(n!)^2(2n+1)!(2n+3)!} \frac{2(n+1)+1}{2n+1} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{2(n+1)+1}{2n+1} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{2n+2+1}{2n+1} \right| ]

[ R = \lim_{n \to \infty} \left| \frac{2n+3}{2n+1} \right| ]

[ R = 1 ]

So, the power series for ( f(x) = \arcsin(x^3) ) converges for ( |x| < 1 ), and the radius of convergence is ( R = 1 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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