In the past few lessons we have focused on power series, and how we can represent functions as a power series. Specifically, we were representing functions similar to the form of the sum of a geometric series as a power series. However, this is not the only way to represent a function as a power ...

In the previous lesson we saw how functions that are similar to the form of the sum of a geometric series can be represented with a power series. However, we can extend this idea to more functions. Specifically, we can represent functions whose derivative or integral is similar to that form of a ...

So far we have seen that power series create functions that are defined for a domain of x values for which the power series is convergent. However, can we actually take a specific function and express it as a power series? Well, there's actually several methods for rewriting functions as power se...

Previously we established that a power series creates a function that is defined for the values of x for which the power series is convergent. Well since a power series is a function, we are naturally interested in knowing its derivative and integral (antiderivative). To find each of these functi...

A power series is a special type of series that involves a variable, usually x, raised to some power of n. As such, each of the terms of a power series involve x, and as a result, form a function f(x). However, this function is not any normal function, it is a function that is only defined for th...

One more method we can use to test for the convergence or divergence of a series is what is known as the Root Test. This test will be particularly useful for series that involve powers of n.

The Root Test involves evaluating the limit of the nth root of the absolute value of the nth term of a se...

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...