# Power Series

1. Nov 2, 2006

### sherlockjones

Could you check over my work on these problems:

1. For what values of $$x$$ is the series $$\sum_{n=0}^{\infty} n!x^{n}$$ convergent?

I used the ratio test: $$\lim_{n\rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| = \lim_{n\rightarrow \infty}(n+1)|x| = \infty$$. So it diverges by the ratio test, but by the definition of the power series it converges at $$x = 0$$

2. For what values of $$x$$ does the series $$\sum_{n=1}^{\infty} \frac{(x-3)^{n}}{n}$$ converge? I again used the ratio test and came up with the following:

$$\lim_{n\rightarrow \infty} \frac{1}{1+\frac{1}{n}}|x-3| = |x-3|$$. Therefore the series diverges when $$|x-3|< 1$$ or when $$2 < x < 4$$. Testing the endpoints, it converges when $$2\leq x< 4$$

3. Find the radius of convergence and the interval of convergence of the series $$\sum_{n=0}^{\infty} \frac{n(x+2)^{n}}{3^{n+1}}$$. Using the ratio test I came up with $$\frac{|x+2|}{3}$$

Thus $$|x+2|<3 \rightarrow R = 3$$. Testing at the endpoints, the interval of convergence is $$(-5, 1)$$

Thanks

Last edited: Nov 2, 2006
2. Nov 2, 2006

### sherlockjones

delete another thread of the same title please

3. Nov 2, 2006

Looks good.

4. Nov 2, 2006

### sherlockjones

Thanks. One more question:

A function $$f$$ is defined by $$f(x) = 1+2x+x^{2}+2x^{3}+x^{4}+...$$. Find the interval of convergence, and an explicit formula for $$f(x)$$.

So $$c_{2n} = 1, c_{2n+1} = 1$$. So $$f(x) = \sum_{n=0}^{\infty} x^{2n} + 2x^{2n+1}$$. Then just apply the ratio test to this to find the interval of convergence?

Last edited: Nov 2, 2006
5. Nov 2, 2006

### Hurkyl

Staff Emeritus
A minor detail: you should indicate where the ratio test says it diverges.

This is a job for the n-th root test! (In fact, once you're used to it, it's almost always at least as good as the ratio test)

I think what you've done is okay... regrouping terms is always a tricky business, but I can you can apply these theorems:

(1) A power series converges absolutely in the interior of its interval of convergence.

(2) You can rearrange / regroup the terms of an absolutely convergent sequence in any way you want.

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