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This is the excellent video on Taylor series that I kept pestering you to watch in Section 9.9. Given an infinitely differentiable function *f*, this video describes how to construct polynomials that approach *f*.

This short video gives a good way to think about why *e^*iπ equals -1. It of course involves imaginary and so leaves the realm of *real* analysis, but not in any way that you can't handle. A second video on this topic by Mathologer is __here__.

This equation was deduced in the final section of the book as a consequence of Taylor series.

This equation was deduced in the final section of the book as a consequence of Taylor series.

The Riemann Hypothesis is one of the great unsolved problems in mathematics. It was mentioned on page 53 of the text, but not stated or discussed. This is an excellent video discussing it.

The Heat Equation is currently an important and active research topic in analysis. This video is a great introduction to this field.

Check out this video for a surprising appearance of π in a physics problem. Throughout the discussion you'll notice several tools discussed in real analysis.

A number is *algebraic* if it is the root of some polynomial with integer coefficients, and it is *transcendental* if it is not. Showing a number is irrational is one thing, but showing it is transcendental is often much more difficult. This video discusses how one might try to prove that *e* and π are transcendental.

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