Difficult geometric series $\sum\limits_{n=1}^{\infty} \frac{\pi^{7n}} {e^{8n}}$

Question

I need to determine the convergence of $$\sum_{n=1}^{\infty} \frac{\pi^{7n}} {e^{8n}}$$ I assume it's a geometric series, but I'm having trouble finding the distinct c and r values. How do I determine this partial sum?

Answer 1

the common ratio in your series is $$r = \pi^7/e^8 \approx 1.01319550322616243138169834655 > 1,$$ so the series diverges. This of course relies on a numeric computation. The proximity of this ratio to $1$ suggests that an analytic approach to showing $r > 1$ is not simple, and is unlikely to be in the scope of this question (i.e., such an argument would be significantly more sophisticated than the original question).

If you accept that $\pi > 3.14$ and $e < 2.72$, then we can compute $$r > \frac{(3.14)^7}{(2.72)^8} = 100 \cdot \frac{314^7}{272^8} = 50 \cdot \frac{157^7}{136^8} = \frac{58781081938437325}{58516894675632128} > 1,$$ in principle computable without relying on a calculator.