- #1
Euler2718
- 90
- 3
Homework Statement
Determine whether each of the following series is convergent or divergent. If the series is convergent, find its sum
[tex] \sum_{i=1}^{\infty} \frac{6}{9i^{2}+6i-8} [/tex]
Homework Equations
Partial fraction decomposition
[tex] \frac{1}{3i-2} - \frac{1}{3i+4} [/tex]
The Attempt at a Solution
The divergence test is inconclusive, so I wrote as partial fractions and started analysing the nth sum:
[tex] S_{n} = \left( 1-\frac{1}{7} \right) + \left( \frac{1}{4} - \frac{1}{10} \right) + \left( \frac{1}{7} - \frac{1}{13} \right) + \left( \frac{1}{10} - \frac{1}{16} \right) + \dots [/tex]
1 and 1/4 are the only terms that do not cancel, but how do I show this in the nth case? I'm having trouble writing it generally.