Calculus II/Test 3

8.1 Infinite sequences and series: Sequences

Read Examples 1, 2, 3 page 554 for understanding.

*  Examples 4, 5 pp557-8 15 February 2017 (UTC)

8.2 Infinite sequences and series: Partial sums

*  Examples 1, 3, 5 p565-568

*  A derivation of SN = ΣN−1j=0  xj   = (1-xN)/(1-x) will certainly be on the test (note error in previous version). If you don't do well on the rest of the test, I will grade it carefully, so don't make any mistakes? The step at the bottom of page 566 of your textbook is breathtakingly beautiful. Also, when can you get a finite result in the limit as N goes to infinity? You will need to know that series and the fact that it is called a w:power series.

We will carefully read Examples 4 and 7, but I don't see a good exam questions for them. Let us replace example 4 by a simpler one. See Talk:Sequences_and_series

Skim three sections

Chapters 8.3 Infinite sequences and series: Integral and comparison tests

Riemann Sum
• Example 1 of 8.3 (p. 577) is instructive, though the integral is too tricky for an exam 23 February 2017

*  Know how to do the integral test on page 577 (see examples 1, 2, 3). I will give you a different integral, and there will be plenty of partial credit for setting up the integral without solving the integral. This is an essential skill because of the link to the Riemann Sum: ${\displaystyle \sum f_{n}\approx \int f(x)dx}$ if the function is smooth enough that Δx=1 is a reasonable approximation (i.e., the rectangles in the figure can have unit thickness). (See review of ${\displaystyle \int _{1}^{\infty }{\frac {dx}{x^{p}}}}$ below)

Chapters 8.4 Infinite sequences and series: Other convergence tests

• The alternating series test is intuitively simple: If the points hop back and forth by smaller and smaller amounts, the it converges. p598
• Therem 1 about absolute convergence on page 588 is important, but is not used much by engineers.
• The Ratio test of 589 is useful for knowing when a Power series converges absolutely.

Chapters 8.5 Infinite sequences and series: Power series

!!!!! Examples 4 and 5 page 596' 18:20, 20 March 2017 (UTC)

• You need to know what a power series is, see see page 592

8.6 Infinite sequences and series: Representations of functions as power series

*  Examples 1,5 pp.601.        20-22 February 2017

• Example 6 was interesting, but I messed up the graph ln(1+x) on the board and misued the log function on Excel. Will not be on test
• We will not do example 7 because it seems so hard to understand: (play double time). Also search for "worry" halfway down this page. The best way to remember this is to use these triangles..

8.7 Infinite sequences and series:Taylor and Maclaurin Series

Womething like this will be on testCarefully study the Taylor-Maclurin Series at p604. Sample problem: Find the 538-th derivative of 13x2122

You will certainly be asked to carefully derive the Taylor series about x=0 for either sine, cosine, or exp (ex), or perhaps ${\displaystyle {\sqrt {1+x}}}$. We will attempt to do (1+x)k, as this was first done by Newton. This is done on Example 8, page 611).

steps to derive Taylor Series for sine

First we do it for a = 0, and show why this makes sense:If the series expansion is true then the following is true

${\displaystyle c_{n}={\frac {f^{(n)}(0)}{n!}}}$

First applicatin is to the Taylor expansion for sin(x):

${\displaystyle \sin(0)=0}$
${\displaystyle {\frac {d}{dx}}\sin(0)=\cos(0)=1}$
${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin(0)=-\sin(0)=0}$
${\displaystyle {\frac {d^{3}}{dx^{3}}}\sin(0)=-\cos(0)=-1}$
${\displaystyle {\frac {d^{4}}{dx^{4}}}\sin(0)=\sin(0)=0}$

Note that this pattern repeats itself because for any n,

${\displaystyle {\frac {d^{n+4}}{dx^{n+4}}}\sin(\theta )={\frac {d^{n}}{dx^{n}}}\sin(\theta )}$
From Wikipedia's Binomial theorem
${\displaystyle {r \choose k}={\frac {r\,(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},}$

According to a hidden comment in w:Special:Permalink/766761934#Newton.27s_generalized_binomial_theorem, we cannot write this as ${\displaystyle {\frac {r!}{k!(r-k)!}}}$ because the definition of ! does not hold for negative numbers. See also w:Gamma function

Time permitting: Section 8.7 examples 6,7,8, 12 pages 610-615. (Example 8 was done previously, and example 12 is extremely useful)

• Before Test 3 I want to carefully review ${\displaystyle \int _{1}^{\infty }{\frac {dx}{x^{p}}}}$ for ${\displaystyle p\neq 1}$. See this 6-min Youtube and/or this excerpt from Wikibooks:
Show ${\displaystyle \int \limits _{1}^{\infty }{\frac {dx}{x^{p}}}={\begin{cases}{\frac {1}{p-1}},&{\text{if }}p>1\\{\text{diverges}},&{\text{if }}p\leq 1\end{cases}}}$
If ${\displaystyle p\neq 1}$ then
${\displaystyle \int \limits _{1}^{\infty }{\frac {dx}{x^{p}}}}$     ${\displaystyle =\lim _{b\to \infty }\int \limits _{1}^{b}x^{-p}dx}$
${\displaystyle =\lim _{b\to \infty }\left({\frac {x^{-p+1}}{-p+1}}\right){\Bigg |}_{1}^{b}=-{\frac {1}{1-p}}\lim _{b\to \infty }\left(b^{-p+1}-1\right)}$
${\displaystyle ={\begin{cases}{\frac {1}{p-1}},&{\text{if }}p>1\\{\text{diverges}},&{\text{if }}p<1\end{cases}}}$
Notice that we had to assume that ${\displaystyle p\neq 1}$ to avoid dividing by 0 (which leads to the natural logarithm).