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 \begin{center}
 {\LARGE \bf   Tests for Convergence of a Series}
 \end{center}
 \vspace{.4in}
 
\begin{defn}[Convergence]   An infinite series $\displaystyle \sum_{i=1}^{\infty} = a_1 + a_2 + a_3 + \cdots + a_n + \cdots $ {\bf converges  to  L} if the sequence of partial sums
\[ s_n = a_1 + a_2 + \cdots a_n\]
converges to a limit $L$. 
\end{defn}
 This definition says that as we add the terms in the infinite string above, the answer gets closer and closer to L, and does not ``pop around''.

 
 \begin{test}[Zero Test]
 If the series $\displaystyle \sum_{i=1}^{\infty} a_i$ converges, then the terms $a_i \rightarrow 0$.
 \end{test}
 \begin{use} {\em The test says that if the terms $a_i$ {\em do not go to zero}, then there is {\bf no way} for the series of partial sums to converge.  Done. Does NOT converge.}
 \end{use}
 
 
 \begin{test}[Integral Test]
 Let $a_i = f(i)$, where f(x) is a continuous function with $ f(x) > 0$, and is decreasing.  Then
 \[{\rm  the \; series }\;\; \sum_{i=1}^{\infty} a_i \; \;{\rm  {\bf converges}\; if\; the \;improper \; integral}\;\; \int_1^{\infty} f(x) dx < \infty.\]
  \[{\rm  the \; series }\;\; \sum_{i=1}^{\infty} a_i \; \;{\rm  {\bf diverges}\; if\; the \;improper \; integral}\;\; \int_1^{\infty} f(x) dx = \infty.\]
 \end{test}
 \begin{use}
 {\rm One application is the convergence of the ``p-series'':}
  \[\sum_{n=1}^{\infty} {1\over{n^p}} \;\;\;{\rm {\bf  converges}\; if } \;\;p > 1, {\rm and\;\;\; {\bf diverges}\; if}\;\; p \leq 1\] 
 \end{use}
 
 \begin{test}[Comparison Test]
 Suppose that $\sum_1^{\infty} a_i$ and $\sum_1^{\infty}$ are series with all terms positive - so $a_i  \geq 0$ and $b_i \geq 0$. 
 \[   \sum_{i=1}^{\infty} b_i \;\;{\rm is\;   {\bf convergent}\; and }\;\;a_i \leq b_i \;\; {\rm for \; all \; i\; \Longrightarrow \;}\;\;\; \sum_{i=1}^{\infty} a_i \;\;\; {\rm is \;   {\bf convergent}.}\]
  \[   \sum_{i=1}^{\infty} b_i \;\;{\rm is\;   {\bf divergent}\; and }\;\;a_i \geq b_i \;\; {\rm for \; all \; i\; \Longrightarrow  \;}\;\;\; \sum_{i=1}^{\infty} a_i \;\;\; {\rm is \;  {\bf  divergent}.}\]
\end{test}
 \begin{use}
 {\rm This is the ``squeeze test'' for infinite series.  Use it to justify the ``cover-up'' method of guessing whether a series converges or diverges.}
 \end{use}
  \newpage
  
 \begin{test}[Limit Comparison Test]
 Suppose that $\displaystyle \sum_1^{\infty} a_i$ and $\displaystyle \sum_1^{\infty}  b_i$ are series with all terms positive. 
  \[   \lim_{i \rightarrow \infty} {{a_i}\over{b_i}} = c > 0 \;\;\; \Longrightarrow \;\;\;     \sum_{i=1}^{\infty} a_i \;\;\; {\rm and} \;\;\sum_{i=1}^{\infty} b_i \;\; {\rm either \; {\bf both\; converge,}\; or \;{\bf both\; diverge}.}\]
  \[   \lim_{i \rightarrow \infty} {{a_i}\over{b_i}} =   0 \;\;\; {\rm and} \;\; \sum_{i=1}^{\infty} a_i \;\; {\rm {\bf converges}} \;\; \Longrightarrow \;\;\; {\rm the\; series}\;\;     \;\;\sum_{i=1}^{\infty} b_i   \;\;\; {\rm {\bf  converges}.}\]
\[   \lim_{i \rightarrow \infty} {{a_i}\over{b_i}} =   \infty \;\;\; {\rm and} \;\; \sum_{i=1}^{\infty} b_i \;\; {\rm {\bf diverges}} \;\; \Longrightarrow \;\;\; {\rm the\; series}\;\;     \;\;\sum_{i=1}^{\infty} a_i   \;\;\; {\rm {\bf  diverges}.}\]
 \end{test}
 \begin{use}{\rm  This is one of the most powerful tests, because it squeezes the two series ``in the limit''.  Just be sure to use it right!  Part one is clear, but {\bf don't mix up the second and third parts}.}
 \end{use}
  
 
 
 
 \begin{test}[Alternating Series Test]
 For the {\rm alternating series} - where all $a_i >0$ 
 $$ \sum_{i=1}^{\infty} (-1)^i a_i \;\;=\;\; a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + \cdots $$
      $a_{i} \geq a_{i+1}$ for all i 
 {\bf and}   $\displaystyle \lim_{i \rightarrow \infty} a_i = 0\;\;\;$
  $\displaystyle \Longrightarrow\;\;\;$   $\displaystyle \sum_i^{\infty} (-1)^i a_i$ {\bf converges}.
 \end{test}
 
 \vspace{.2in} 
 
 \begin{defn}[Absolute Convergence]
   \[\sum_i^{\infty} a_i \;\;{\rm is\; {\bf absolutely\; convergent}} \;\;\Longleftrightarrow \;\;{\rm the \;sum\; of\; {\bf absolute\; values\;}} \;\;\sum_i^{\infty} |a_i| \;\;{\rm  is\; convergent}.\]
 \end{defn}

 \vspace{.2in} 
 
 
 
 \begin{test}[Ratio Test]
  \[   \lim_{i \rightarrow \infty} {{a_{i+1}}\over{a_i}} =   L < 1 \;\;\;   \Longrightarrow     \;\;\sum_{i=1}^{\infty} |a_i|   \;\;\; {\rm {\bf  converges} } \;\;\;   \Longrightarrow     \;\;\sum_{i=1}^{\infty} a_i   \;\;\; {\rm {\bf  converges}}.  \]
 \[   \lim_{i \rightarrow \infty} {{a_{i+1}}\over{a_i}} =   L > 1 \;\;\;   \Longrightarrow           \;\;\sum_{i=1}^{\infty} a_i   \;\;\; {\rm {\bf  diverges}}.  \]
\end{test}
 
 
 \begin{test}[Root Test]
  \[   \lim_{n \rightarrow \infty} \sqrt[n]{a_n} =   L < 1 \;\;\;   \Longrightarrow     \;\;\sum_{n=1}^{\infty} |a_n|   \;\;\; {\rm {\bf  converges} } \;\;\;   \Longrightarrow     \;\;\sum_{n=1}^{\infty} a_n   \;\;\; {\rm {\bf  converges}}.  \]
 \[   \lim_{n \rightarrow \infty}  \sqrt[n]{a_n} =   L > 1 \;\;\;   \Longrightarrow           \;\;\sum_{n=1}^{\infty} a_n   \;\;\; {\rm {\bf  diverges}}.  \]
\end{test}

 
 
 
 
 
 
 
 
 
 
 
 
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