Math Analysis Week 5

Chapter 19

So we have two weeks to cover chapter 19. If we make it to 19.21 this week I think we are on schedule. I posted up to

19.1

This section is an intro to series.
Note and understand the last paragraph of page 568 ( before the exercises).

  • Work problems 1, try all, (a) -(h)), 2, 3, 4

  • Problem #5 looks hard, but it might not be. GIve it a try

19.1 Taylor’s Series

Apparently they refer to Taylor’s Formula (4.3-8) and we have not worked this out. If you don’t remember where the expression

\[f(x) = f(a)+f'(a)(x-a)+\ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n +R_{n+1}\]

comes from, assume \(f(x)\) can be written as a power series centered at zero (an infinite polynomial), that is \[f(x) = a_0 + a_1 x + a_2x^2+\ldots a_nx^n+\ldots\] and

  1. evaluate both sides at \(x=0\).

  2. this give you \(a_0\)

  3. take derivative of both sides,

  4. evaluate both sides at \(x=0\)

  5. this gives you \(a_1\)

  6. take the second derivative on both sides

  7. evaluate both sides at \(x=0\)

  8. this gives you \(a_2\)

  9. etc...

You should become very familiar and comfortable with the taylor series of \(e^x, \cos x, \sin x\), which is problem #1

19.11 Inverse Tangent

This is a cool way to find a power series (in fact, the Taylor SEries ) for \(\tan^{-1}(x)\), and use it to estimate \(\frac{pi}{4}\) as an infinte sum (19.11-3).

I particularly don’t like the problems in this mini-section. But if you have extra time, you can have fun trying them out.

19.2 Series of positive terms

Why do they call them “non-negative” instead of “positive”? I have no idea!

Theorems II and III are known as a comparison tests in most Calculus Textbooks. In case you come across this while tutoring students.

The problems in this section are typical problems on this material and I would like us to do them all. but let’s settle for doing all odds for this week. ( If you find that a particular odd problem seems too hard, do the following even problem instead).

19.21 The integral test

After understanding the proofs ob both, theorem IV and the estimate 19.21-3, work the problems

  • 1 (a)- (h)

  • 3, 4, 5

19.22 Ratio tests

In our calculus textbooks this test is the most useful to determine if a series converges or not. Check both proofs (Theorem V and Theorem VI) and ..all problems!