The Ratio Test involves finding the common ratio r by taking the absolute value of the difference between consecutive terms divided by the previous term (i.e., |a_n+1/a_n|). We then calculate lim n->infinity r^n, which is called the limit superior or “lim sup” for short. If this limit is less than 1, then your series converges! If it’s greater than 1, then it diverges (sorry buddy). And if it’s equal to 1, well…you might have a problem on your hands.

Let me give you an example: let’s say we want to check whether the series n=0(3/4)^n converges or not using our trusty Ratio Test friend. First, we calculate r as follows: |a_n+1/a_n| = |(3/4)^{n+1}/(3/4)^n| = (3/4). Since this value is less than 1, we can take the limit superior and see if it’s also less than 1. The limit superior is: lim n->infinity r^n = lim n->infinity (3/4)^n = 0. So our series converges!

The Root Test involves looking at the nth root of each term (i.e., |a_n|^(1/n)). If this value approaches 0 as n goes to infinity, then your series converges! Let’s say we want to check whether the series n=0(2^(-3*n)) converges or not using our Root Test friend. First, we calculate |a_n|^(1/n) for each term: (2^-3)^(1/n). Since this value approaches 0 as n goes to infinity, our series converges!

These guys are super useful when you’re dealing with complex math problems that involve infinite sums (which can be pretty ***** impossible to calculate). Just remember: if your limit superior is less than 1, then your series converges! If it’s greater than 1, then it diverges…and if it’s equal to 1, well…you might have a problem on your hands.

In terms of rates of convergence, the Ratio Test can also provide us with information about how quickly or slowly our series is converging. If r is very close to 1 (but less than 1), then our series will be convergent but may take a long time to do so. On the other hand, if r is much smaller than 1, then our series will converge more quickly.