The comparison test limit

 The objectives of this post are: 

a) Define the limit comparison test

2) Use it to find the convergence or divergence of a series

The comparison test limit works effectively if it's possible to find another series that  satisfies the hypothesis of the test. Let's consider the series:

Let's compare it to the series:

 

This series is convergent a a p-series: p = 2>.1. Let's compare the general terms of both series.. We have:

1/n²-1>1/n²

We can't say anything about the given series meaning we can't say if it's convergent or divergent. In order for the comparison test to work here we should have 1/n²-1<1/n².

Let's find a theory to solve this problem. 

These results can be summarized in the following theorem:

Examples

For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test doesn't apply, say so.