009C Sample Midterm 1, Problem 3

Determine whether the following series converges absolutely, conditionally or whether it diverges.

Be sure to justify your answers!

$\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}$

Foundations:
1. A series $\sum a_{n}$ is absolutely convergent if
the series $\sum |a_{n}|$ converges.
2. A series $\sum a_{n}$ is conditionally convergent if
the series $\sum |a_{n}|$ diverges and
the series $\sum a_{n}$ converges.

Solution:

Step 1:
First, we take the absolute value of the terms in the original series.
Let $a_{n}={\frac {(-1)^{n}}{n}}.$
Therefore,
${\begin{array}{rcl}\displaystyle {\sum _{n=1}^{\infty }|a_{n}|}&=&\displaystyle {\sum _{n=1}^{\infty }{\bigg |}{\frac {(-1)^{n}}{n}}{\bigg |}}\\&&\\&=&\displaystyle {\sum _{n=1}^{\infty }{\frac {1}{n}}.}\end{array}}$

Step 2:
This series is the harmonic series (or $p$-series with $p=1$).
So, it diverges. Hence the series
$\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}$
is not absolutely convergent.

Step 3:
Now, we need to look back at the original series to see
if it is conditionally converges.

Step 4:

Final Answer:

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