Integral calculus

Would you believe me if I told you that if you walked straight at a wall that you would never actually get to the wall? Integral calculus allows you to mathematically prove this crazy idea. When you think of calculus, think tiny as in infinitesimal. By subdividing the space between you and the wall into ever smaller divisions, you can mathematically establish that there is an infinite number of divisions, and you can never actually get to the wall. Do not try this at home kids, not without some help from integrals and derivatives, the basic tools of calculus. The study of integral calculus includes: integrals and their inverse, differentials, derivatives, anti-derivatives, and approximating the area of curvilinear regions.
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Sequences, series, and function approximation

Sequences, series and approximating functions. Maclaurin and Taylor series.
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All content in “Sequences, series, and function approximation”

Sequences

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

Sequence convergence and divergence

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?).

Series

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.

Geometric series

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!

Tests for convergence and divergence

We will now deepen our convergence and divergence tool kits by exploring a series of "tests" we can apply to determine the behavior of some series.

Estimating infinite series

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 also 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).

Power series function representation using algebra

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 (and the special case of a geometric series). In later tutorials, we'll use calculus to find the power series of more types of functions.

Maclaurin series and Euler's identity

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.

Taylor series approximations

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.

Sal's old Maclaurin and Taylor series tutorial

Everything in this tutorial is covered (with better resolution and handwriting) in the "other" Maclaurin and Taylor series tutorial, but this one has a bit of old-school charm so we are keeping it here for historical reasons.