- #1
danielbaker453
- 26
- 2
Homework Statement
Test the series for convergence or divergence
##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...##
Homework Equations
rn=abs(an+1/an)
The Attempt at a Solution
With some effort I was able to figure out the 'n' th tern of the series
an =
\begin{cases}
2^{-(0.5n+1.5)} & \text{if } n is odd \\
-3^{-(0.5n+1)} & \text{if } n is even
\end{cases}
Next, I considered two cases,
Case I. When an is odd,
This gives rn={(2/3)^(0.5n+1.5)}<1 for all n
Thus the terms of the series seem to decrease.
As n----->infinity, rn----->0
The series appears to be convergent.
Case II. When an is even
rn=0.5{(3/2)^(0.5n+1)}>1
As n------>infinity, rn----->infinity
This time around the series seems to be divergent.
What am I doing wrong? What should I do to properly determine the convergence/divergence of the series?