I know how to find the interval of convergence, but I don't understand when to check for convergence at the endpoints (to see if I should include it or not)
Could someone help me understand this?
Thanks!
Of course! Knowing when to check for convergence at the endpoints is an important part of finding the interval of convergence. To determine whether to include the endpoints in the interval or not, you can follow these steps:
1. Find the interval of convergence using a convergence test such as the ratio test, root test, or the alternating series test. This will give you a range of values for which the series converges.
2. Once you have identified the interval of convergence, you need to check if the series converges at the endpoints of the interval. To do this, substitute the endpoints into the original series and see if the resulting series either converges or diverges.
3. If the series converges at one or both endpoints, then you should include them in the interval of convergence. This means that the series converges for all values within the interval including the endpoints.
4. However, if the series diverges at one or both endpoints, then you should exclude them from the interval of convergence. This means that the series only converges for values within the interval, excluding the endpoints.
By checking for convergence at the endpoints, you can determine whether to include or exclude them from the final interval of convergence. Remember that the convergence or divergence at the endpoints can be determined by substituting the endpoints into the original series and examining the behavior of the resulting series.