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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Problem Set 5, {\it Due:}  10/14}
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I.  Decide whether or not each of the following infinite series
converges, and justify your claim using one of the results
in sections 2.4 and 2.7 of Abbott:
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\item{A)} $$\sum_{n=0}^\infty {1\over \sqrt{2n+1}}$$

\item{B)} $$\sum_{n=0}^\infty {1\over (2n+1)^2}$$

\item{C)} $$\sum_{n=0}^\infty (-1)^n {2^n \over 3^{2n}}$$

\item{D)} $$\sum_{n=0}^\infty (-1)^n {n^3 + n\over 2n^3 + n + 1}$$

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II.  
\item{A)} For which $x \in [0,2\pi]$ does the series
$$\sum_{n=1}^\infty \sin^n(x)$$ converge?  What is the sum for those $x$?

\item{B)} For which $x \in {\bf R}$ does the series
$$\sum_{n=1}^\infty e^{nx}$$ converge? What is the sum for those $x$?


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III. From Abbott: 2.7.3, 2.7.4, 2.7.9.

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IV.  Use the Ratio Test (Exercise 2.7.9 above) to determine for
which $x \in {\bf R}$ the series 
$$\sum_{n=0}^\infty {(x-1)^n \over 3^n}$$ 
converges.
(Hint: The Ratio Test will give an open interval of values 
of $x$ on which the series converges absolutely.  What
happens at the endpoints of that interval?)
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