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\begin{center}
\textbf{Power Series}
\end{center}

\noindent
Definition.  Suppose that $\{a_i\}_{i=1}^\infty$ is a sequence.  Then
\begin{eqnarray*}
\sum_{i=1}^\infty a_i & = & L
\end{eqnarray*}
means that the sequence of partial sums
$$\Big\{\sum_{i=1}^n a_i \Big\}_{n=1}^\infty$$ has sequential limit $L$.  Such a series is said to converge; a series for which no such limit exists is said to diverge.

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Theorem 5.1. Suppose that the series $\sum_{n=0}^{\infty} a_n$ converges.
Then
$$\lim_{n \rightarrow \infty} |a_n| = 0.$$

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Exercise.  Show that the converse to Theorem 5.2 is not true.

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Theorem 5.2.  Let $r$ be a number then, the series
$$\sum_{n=0}^{\infty} r^n$$
converges if and only if $|r| < 1.$  Furthermore, if $|r| < 1$ then
$$\sum_{n=0}^{\infty} r^n= \frac{1}{1-r}.$$

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Theorem 5.3. If the series $\sum_{n=0}^{\infty} |a_n|$ converges, then so does $\sum_{n=0}^{\infty} a_n$.

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Exercise.  Show that the converse to Theorem 5.3 is not true.

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Definition.  $\int_K^{\infty} fdg$ means the following limit if it exists: $$\lim_{n\rightarrow \infty} \int_K^{n} fdg.$$

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Theorem 5.4 [The integral test]. Suppose that the function $f$ is defined for all positive integers and that $f|_{ [K, \infty)}$ is defined and is positive and decreasing.  Then $\sum_{n=1}^{\infty}f(n)$ converges if and only if $\int_K^{\infty} f$ exists.

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Theorem 5.5 [The comparison test].  Suppose that $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ are two sequences so that $0 \le a_n \le b_n$ for all $n \in \mathbb{N}$.  Then:

\qquad 1.) If $\sum_{n=1}^{\infty} b_n$ converges, then so does $\sum_{n=1}^{\infty} a_n$;

\qquad 1.) If $\sum_{n=1}^{\infty} a_n$ diverges, then so does $\sum_{n=1}^{\infty} b_n$.

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Theorem 5.6 [The limit comparison test]. Suppose that each of $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ is a series of positive numbers and that
$$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L \ne 0.$$
Then $\sum_{n=1}^{\infty}a_n$ converges if and only if $\sum_{n=1}^{\infty}b_n$ converges.

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Exercise.  If $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 0$ then something still can be said about the relationship between the two series.  What is that?

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Theorem 5.7 [The ratio test]. Consider the series $\sum_{n=0}^{\infty} a_n$ and let
$$L = \lim_{n \rightarrow \infty} \Big{|} \frac{a_{n+1}}{a_n}\Big{|}.$$ Then:
\begin{eqnarray*}
\mbox{If} & L<1 & \mbox{then the series converges} \\
\mbox{If} & L>1 & \mbox{then the series diverges}.
\end{eqnarray*}

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Exercise.  Let $\sum_{n=0}^{\infty} a_n$ be a series so that $a_n > 0$ and suppose
$$\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = 1.$$

(i.) Find an example where the series $\sum_{n=0}^{\infty} a_n$ converges;

(ii.) Find an example where the series $\sum_{n=0}^{\infty} a_n$ diverges.


\newpage

\begin{center}
\textbf{Power Series}
\end{center}

Theorem 5.8.  Suppose that $\{A_n\}_{n=1}^{\infty}$ is a sequence of numbers and
$$\lim_{n \rightarrow \infty}  \Big{|} \frac{A_n}{A_{n+1}} \Big{|} = r.$$ Then if $|x| < r$ the series $\sum_{n=1}^{\infty} A_n x^n$ converges.

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For the following theorems assume that $\{A_n\}_{n=1}^{\infty}$ is a sequence of numbers, $r<1$ is a number so that
$$\lim_{n \rightarrow \infty}  \Big{|} \frac{A_{n}}{A_{n+1}} \Big{|} = r$$ and $f$ is defined by
$$f(x) = \sum_{n=0}^{\infty} A_n x^n \mbox{ for } -r < x < r.$$


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Theorem 5.9.  If $0< \delta < r$, then then sequence of functions $f_n = \sum_{i=0}^{n}A_i x^i$ converges uniformly to the function $f$ on the interval $[-\delta, \delta]$.

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Theorem 5.10.  If $0< x < r$, then
$$\int_0^x f(t) dt = \sum_{n=0}^{\infty}A_n \frac{ x^{n+1}}{n+1}.$$

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Theorem 5.11.  If $0< x < r$, then
$$f'(x) = \sum_{n=1}^\infty A_n n x^{n-1}.$$


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