009C Sample Midterm 2, Problem 2

Determine convergence or divergence:

$\sum _{n=1}^{\infty }{\frac {3^{n}}{n}}$

Foundations:
Direct Comparison Test
Let $\{a_{n}\}$ and $\{b_{n}\}$ be positive sequences where $a_{n}\leq b_{n}$
for all $n\geq N$ for some $N\geq 1.$
1. If $\sum _{n=1}^{\infty }b_{n}$ converges, then $\sum _{n=1}^{\infty }a_{n}$ converges.
2. If $\sum _{n=1}^{\infty }a_{n}$ diverges, then $\sum _{n=1}^{\infty }b_{n}$ diverges.

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 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 p} -series with 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 p=1.} )
Hence, 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 b_n} diverges.

Step 3:
Also, we have 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 b_n<a_n} since
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 \frac{1}{n}<\frac{3^n}{n}}
for all 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 n\ge 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

Return to Sample Exam