I Hope this might help u to read better,
assess the series
infinity
sum of ((-2)^n)/(3n+1)!
n = 0
Use the generalized ratio test,
Note that,
Thus,
as
Which is less than 1 thus the series converges.
---
Also, the sequence,
Fits the conditions for Leibniz's alternating series test, thus it absolutely convergent.
Just analytically thinking about this, I can see that the factorial will very strongly outgrow the exponential as x gets larger. But more formally...
Thus this diverges.
Another way is to just take the limit of the original series.
If this limit is anything but zero, the series diverges. If it is zero, you must use another test to help you.