# Sequences, series, and function approximation

Contents

Now we switch gears away from integration to talk about sequences and series. Much of calculus is about dealing with infinity, and this topic has us dancing very closely with infinity itself. Sequences are infinite lists, series are infinite sums, and there is no small amount of delicacy involved in managing these two objects.

In this tutorial, we'll review what sequences are, associated notation and convergence/divergence of sequences.

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?).

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.

Whether you are computing mortgage payments or calculating the distance traveled by a bouncing ball, geometric series show up in life far more than you imagine. This tutorial will review all the important concepts and more!

Here you will learn several "tests" which determine whether certain series converge or diverge.

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).

Now that we're familiar with the idea of an infinite series, we can now think about functions that are defined using infinite series. In particular, we'll begin to look at the power series representation of a function, which are kind of like infinite polynomials. In later tutorials, we'll use calculus to find expressions of several common functions as power series.

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun.
If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

As we've already seen, Maclaurin series are special cases of Taylor series centered at 0. We'll now focus on more generalized Taylor series.