I am doing a homework problem where we need to use the appropriate comparison test (Direct or limit) to determine if the following series is convergent of divergent:
$$\sum_{i=1}^{\infty} \frac{i!}{5\cdot8\cdot11\cdot\cdot\cdot(3i+2)}$$
Of course, the ratio test would be straight forward for this case. However we have not yet gotten to it. The question specifically states that we must use a comparison test.
My first thought is that this is convergent. The denominator is getting larger than the numerator faster. I would imagine doing a direct comparison test would be the best. I would just need to find a value that is greater than i! but less than the denominator that still results in either geometric series or p series.
Am I taking the wrong approach?
Best Answer
Here is a good way to use the comparison test:
$$\begin{align}5\times8\times11\times\dots\times(3i+2)&>2\times4\times6\times\dots\times(2i)\\&=(2\times1)(2\times2)(2\times3)\dots(2\times i)\\&=2^ii!\end{align}$$
Thus,
$$\frac{i!}{5\cdot8\cdot11\cdot\cdot\cdot(3i+2)}<\frac{i!}{2^ii!}=\frac1{2^i}$$
And this is just a geometric series that converges.