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...

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...

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...

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 n...

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...

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...

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,...

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...