Direct Comparison Test

Another way to determine if a series converges or diverges is with the direct comparison test. With this test, you create a series that you can quickly evaluate that is very similar to the one provided, but it's either greater or less than the provided one, given that both series are positive. If the new series converges and is greater than the original series, the original series converges.If the new series diverges and is less than the original series, the original series diverges.

Steps:
1. Create a similar series to the original series, and determine if it's greater or less than the original series
2. Evaluate the behavior of the new series and see if it converges or diverges
3. Check with the conditions above if you can make a claim about the original series.


Question 1: \[\sum _{n=2}^{\infty }\:{{3+cos(n)} \over n^2}\]

Question 2: \[\sum _{n=2}^{\infty }\:{1 \over ln(n)}\]

Question 3: \[\sum _{n=2}^{\infty }\:{1 \over n^3 +1}\]