You're familiar with sequences and have been eager to sum them up. Well wait no longer! In this tutorial, we'll see that series are just sums of sequences and familiarize ourselves with the notation.
Learn how to find the value of some special classes of series. This material is relatively advanced and isn't included in the AP Calculus course.
Power_series_function_representation
We already saw how Taylor and Maclaurin series can be used to represent a variety of functions. In this tutorial, we will expand the scope of functions that we can represent with power series, by enriching our toolkit!
In this tutorial, we'll review what sequences are, their associated notation, and formulas of sequences.
Taylor_Maclaurin_polynomials_intro
Taylor polynomials are a very clever way of approximating any function with a polynomial. Maclaurin polynomials are a special class of Taylor polynomials. Learn how these polynomials work in this tutorial.
If the sum of a series must be smaller than the sum of another series that converges, then that series must converge as well. Makes sense, no? This notion, and similar ones, are at the basis of the comparison convergence tests.
Now that we understand what a sequence is, we're going to think about what happens to the terms of a sequence at infinity (do they approach 0, a finite value, or +- infinity?).
We've spent a lot of time thinking about whether a series converges or diverges. But, even if we can determine that a series converges, how can we figure out what it converges to? This tutorial will show techniques of estimating what a series converges to, and for determining how good our estimates are. This is super useful because most series can't be precisely evaluated (like we were able to do with infinite geometric series).
This might seem unbelievable at first, but for a certain class of geometric sequences, we can take the sum of all the infinite terms in the sequence and end up with a finite number! This sum is called an infinite geometric series. Learn when and how it can be found.
Partial sums are another way to think of finite series. Why partial? Because they are a part of an infinite series! Thinking of the partial sums of an infinite series helps us analyze the infinite series itself.
A finite geometric series is the sum of the first few terms of a geometric sequence. It turns out there's a quick way of finding such a sum, without having to really sum all the terms one-by-one.
How can we tell whether a series converges or diverges? There is an impressive repository of tools that can help us with this question. In this tutorial, we will cover the basic tests: n-th term test, integral test, and p-series test. These tests can help us later on with the more elaborate convergence tests.
Review you knowledge about series with some challenging exercises.
Maclaurin_series_of_sin_x_cos_x_and_e_
A Maclaurin series is essentially a Maclaurin polynomial with infinite terms. It turns out that the Maclaurin series of sin(x), cos(x), and eˣ return the exact same values as the functions themselves, for any x-value. Wow! Introduce yourselves to these special series and discover one of the most fascinating areas of all of mathematics!
Ratio_alternating_series_tests
The ratio test is one of the most useful tests for series convergence. Alternating series are a special class of series that have their own, very useful test for convergence.
Power series is a sum of terms of the general form aₙ(x-a)ⁿ. As you can see, it's actually a function whose sum depends on the x-value. It is a generalization of geometric series, and an extremely useful class of series. Learn more about it in this tutorial.
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