Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:

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Next: Wang Ying's Solution of Homework Assignment 20 Previous: Class Notes for the Week of February 10 (9 of 9)

Homework Assignment 20

Assigned Tuesday February 24; due Friday March 7, 2PM at SS 1071

this document in PDF: HW20.pdf

Required reading

All of Spivak Chapter 20.

To be handed in

From Spivak Chapter 20: Even parts of 1, 3, 4, 8.

Recommended for extra practice

From Spivak Chapter 20: Odd parts of 1, 3, 4, 8 and all of 6, 9.

Just for fun

According to your trustworthy professor, $$P_{2n+1,0,\sin}(x)=\sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ 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?

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