# Calculus II

## Day 28-Lecture 56

15 May 2017, Monday 9:40 --
**The Last Lecture**

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# Applications and the irrationality of *e*

Here I forgot to carry over the factor $\displaystyle \frac{1}{2}$ to the answer, so the correct answer is
$\displaystyle -\frac{2}{3}$.

The significance of Machin's formula is that the convergence of the Left Hand Side is now extremely fast as opposed to the convergence of arctan* x*.

Google "machin type formulas" to see much faster converging series.

The signficance of transcendental numbers is that there are more transcendental numbers than non-transcendental (algebraic) numbers on the real line, yet it is too complicated to show that a number is transcendental. The proof that *e* is transcendental is however easy enough to be followed by a Calculus background. That is why it is mentioned here. Just Google "the transcendence of e" and read any of the proofs. Similarly you can also follow a proof of the transcendence of π if you accept that square root of negative one exists!

For questions

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