Another test for the convergence or divergence of a series that we can use is what is known as the Ratio Test. This test will be particularly useful for series that involve factorials and exponential parts, and not so helpful for series that are similar to a p-series.

The Ratio Test involves eva...

Previously we saw that if a series has terms that alternate between being positive and negative that we could determine the convergence or divergence of that series by using the alternating series test. However, if we encounter a series that has both negative and positive terms, but does not alte...

Another special type of series you may encounter is what is known as an alternating series. An alternating series is a series whose terms alternate between being positive and negative. This is usually caused by having the expression (-1)^n or (-1)^(n+1) within a series. In either case, both expr...

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 focus of thi...

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 its convergence. Specifical...

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 was...

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 be 2+4+8+16+32+·...

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 actually ...

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...