Page 204 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

and the trigonometric function cosine is equal to

∞ 2n
n
x
cos(x)= (−1) .
(2n)!
n=0
Prove that for each x ∈ R , these series converge.

Finally, using the standard definitions of the exponential function e and the definition of the
x
trigonometric functions sin(x) and cos(x) together with the definition of i , prove Euler’s
equation:

iπ
e + 1=0.

9. What are all the right triangles with integer sides? In other words, what are all integer
solutions to

2
2
2
x + y = z .
Are there infinitely many? Such an x, y, z are called a Pythagorean Triple. *What about integer
n
solutions to x + y = z for n ≥ 3. What are some integer solutions to this equation? Are
n
n
there infinitely many solutions?

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