In the last lesson we looked at the first comparison test for series known as the Direct Comparison Test. Upon introducing that test, I mentioned that there was a second comparison test that we will look at in the next lesson. That test is called the Limit Comparison Test, and is the...
In the various tests for convergence of a series we have looked at so far (geometric series, p-series, and the integral test), the terms of the series needed to be pretty simple and have special characteristics in order for each test to be applied. Any slight deviation from these characteristics...
Another special type of series you may encounter is what is known as a p-series. A p-series is a series that involves n raised to a power p in the denominator of the nth term. This type of series is special because similar to a geometric series, we can easily determine...
When we first introduced series, we looked a quick test that can be performed to determine if it is a diverging series. This test was the Divergence Test, and just involved taking the limit of the nth term of the series. If that limit was not equal to 0, then the series diverged! So this test...
Another special type of series you may encounter is what is known as a geometric series. A geometric series is a series where each subsequent term is found by multiplying the previous term by the same value known as the "common ratio".
A basic example of a geometric series would...
A special type of series you may encounter is what is known as a telescoping series. A telescoping series is a series whose terms collapse, or "telescope." In other words, we would say that many of the terms in the series cancel out, leaving us with only a couple terms to work with that...
An important application of sequences is how they can be used to represent infinite summations, or what we call infinite series, commonly shortened to just "series."
To put it simply, a series is the sum of the terms of a sequence. For example, if we have a sequence represented by the nth term 3...
Another way to decide if a sequence converges or not is by determining if it is monotonic and bounded. If these two conditions are met, we can conclude that a sequence converges without ever needing to use a limit. But what does it mean for a sequence to be monotonic and bounded?
A monotonic...
We are now making our way into the next major group of concepts in Calculus 2, which is sequences and series. We will start by first introducing sequences, which is the least complicated of the two.
In mathematics, a sequence is an ordered list of terms. A very basic example would be a...
In the last lesson we took a brief side trip and revisited limits from Calculus 1. We looked at a new technique known as L'Hopital's rule for evaluating limits involving indeterminate forms. Now while that may have seemed unrelated to our journey thus far through the various advanced...
