Could you check over my work on these problems:(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

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

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

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

Thanks

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# Power Series

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