009C Sample Midterm 2, Problem 2

Determine convergence or divergence:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty \frac{3^n}{n}}

Foundations:
Direct Comparison Test

Solution:

Step 1:
First, we note that
${\frac {3^{n}}{n}}>0$
for all $n\geq 1.$
This means that we can use a comparison test on this series.
Let $a_{n}={\frac {3^{n}}{n}}.$

Step 2:
Let $b_{n}={\frac {1}{n}}.$
We want to compare the series in this problem with
$\sum _{n=1}^{\infty }{\frac {1}{n}}.$
This is the harmonic series (or $p$-series with $p=1.$)
Hence, $\sum _{n=1}^{\infty }b_{n}$ diverges.

Step 3:
Also, we have $b_{n}<a_{n}$ since
${\frac {1}{n}}<{\frac {3^{n}}{n}}$
for all $n\geq 1.$
Therefore, the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty a_n} diverges
by the Direct Comparison Test.

Final Answer:
diverges

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