Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: (84) Next: Brook Taylor
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Homework Assignment 21

Assigned Tuesday March 2; due Friday March 12, 2PM, at SS 1071

This document in PDF: HW.pdf

Required reading. Spivak's Chapter 20.

To be handed in. From Spivak Chapter 20: Odd parts of 1, 3, 4, 8.

Recommended for extra practice. From Spivak Chapter 20: Even parts of 1, 3, 4, 8 and all of 6, 9.

Just for fun. According to your trustworthy professor, $ \displaystyle P_{2n+1,0,\sin}(x)=\sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}$ should approach $ \sin x$ when $ n$ goes to infinity. Here are the first few values of $ P_{2n+1,0,\sin}(157)$:

$ n$ $ P_{2n+1,0,\sin}(157)$
0 157.0
1 -644825.1666
2 794263446.1416
3 -465722259874.7894
4 159244913619814.5429
5 -35629004757275297.7787
6 5619143855101017161.3172
7 -658116552443218272478.0047
8 59490490719826164706638.3418
9 -4275606060900548165855463.4918
10 250142953226934230105633222.4574
100 $ \sim 5.653\cdot 10^{63}$

In widths of hydrogen atoms that last value is way more than the diameter of the observable universe. Yet surely you remember that $ \vert\sin 157\vert\leq 1$; in fact, my computer tells me that $ \sin 157$ is approximately -0.0795485. In the light of that and in the light of the above table, do you still trust your professor?

The Small Print. For $ n=(200,\ 205,\ 210,\ 215,\ 220)$ we get $ P_{2n+1,0,\sin}(157)=(1.8512\cdot 10^8,\ -13102.9,\ 0.648331,\ -0.0795805,\ -0.0795485)$.

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Dror Bar-Natan 2004-03-01