The menu LIMITS AND SERIES Practice using Taylor series

hp calculators HP 50g Using Taylor Series The menu LIMITS AND SERIES The menu LIMITS AND SERIES contains commands related to limits. To access it you press !Ö. You are presented then the calculus menu as a CHOOSE box:

Figure 1

Its second menu item is 2.LIMITS AND SERIES... You can use such CHOOSE boxes much like menus of computer applications. You can move the selection using the arrow keys. You can also jump to a certain menu item by typing the first few letters of the command or the number at the left of the menu item. Pressing the `key or the menu key %OK% will execute the selected item. In this CHOOSE box you press 2 to select 2.LIMITS AND SERIES.. and then ` or %OK% to display the menu:

Figure 2

The command DIVPC needs two polynomials and an integer. It returns the increasing power quotient of the two polynomials up to an order indicated by the integer. The command lim takes an algebraic object and an equation of the form variable=expression. It returns the limit of the algebraic expression when the given variable approaches the expression at the right hand side of the equation. The command SERIES needs an algebraic expression, and equation of the form variable=expression, and an integer. It returns a list at stack level 2 and an equation at stack level 1. The list contains 4 items: The limit of the expression when the given variable approaches the expression at the right hand side of the equation. The equivalent value expression at that point. The power expansion at that point. And finally the order of the residual at that point. The equation on stack level 1 is of the form h=variable-expression, where variable and expression are the same as in the equation variable=expression that we provided as argument for the command. The command TAYLOR0 performs a Maclaurin series expansion of an expression in the default independent variable, VX...

...Gods Gift to Calculators: The TaylorSeries
It is incredible how far calculators have come since my parents were in
college, which was when the square root key came out. Calculators since then
have evolved into machines that can take natural logarithms, sines, cosines,
arcsines, and so on. The funny thing is that calculators have not gotten any
"smarter" since then. In fact, calculators are still basically limited to the
four basic operations: addition, subtraction, multiplication, and division! So
what is it that allows calculators to evaluate logs, trigonometric functions,
and exponents? This ability is due in large part to the Taylorseries, which
has allowed mathematicians (and calculators) to approximate functions,such as
those given above, with polynomials. These polynomials, called Taylor
Polynomials, are easy for a calculator manipulate because the calculator uses
only the four basic arithmetic operators.
So how do mathematicians take a function and turn it into a polynomial
function? Lets find out. First, lets assume that we have a function in the form
y= f(x) that looks like the graph below.
We'll start out trying to approximate function values near x=0. To do
this we start out using the lowest order polynomial, f0(x)=a0, that passes
through the y-intercept of the graph (0,f(0)). So f(0)=ao.
Next, we see that the graph of f1(x)= a0 + a1x will also pass through x=
0, and will...

...(x).
5a.
When y = sin (1), y = 0.841. Using the Taylorseries with two terms, y = 0.830.
When y = sin (5), y = -0.958. Using the Taylorseries with two terms, y = - 15.8.
When y = cos (1), y = 0.540. Using the Taylorseries with two terms, y= 0.500.
When y = cos (5), y = 0.284. Using the Taylorseries with two terms, y = - 11.5.
By using the formula, Percentage Error =
Percentage Error for Taylorseries with two terms =
= 1377.18 % ≈ 1380 % (3sf)
5b.
When y = sin (1), y = 0.841. Using the Taylorseries with three terms, y = 0.842.
When y = sin (5), y = -0.958. Using the Taylorseries with three terms, y = 10.2.
When y = cos (1), y = 0.540. Using the Taylorseries with three terms, y = 0.542.
When y = cos (5), y = 0.284. Using the Taylorseries with three terms, y = 14.5.
Percentage Error for Taylorseries with two terms=
= 1492.80% ≈ 1490% (3sf)
5c.
When y= sin (1), y = 0.841. Using the Taylorseries with four terms, y= 0.841.
When y = sin (5), y = -0.958. Using the Taylorseries with four terms, y= -5.29.
When y = cos (1), y = 0.540. Using the Taylor...

...MATH 152 MIDTERM I 02.11.2012 P1 P2 P3 Name&Surname: Student ID: TOTAL
Instructions. Show all your work. Cell phones are strictly forbidden. Exam Duration : 70 min. 1. Show that 1 p n (ln n) n=2 converges if and only if p > 1: Solution: Apply integral test: Z Z
ln R 1 X
R
2
1 p dx x (ln x) p=1 p 6= 1
let ln (x) = u then
ln 2
so that when p = 1 and p < 1 integral diverges by letting R ! 1, so does the series. When p > 1 then integral converges to ! 1 p 1 p 1 p (ln R) (ln 2) (ln 2) lim = , R!1 1 p 1 p 1 p so does the series. 2. (18 pts.) Find the in…nite sum 1 : n (n + 2) n=1 Solution: See that 1 1 = n (n + 2) n 1 n+2
1 X
8 R < ln ujln 2 ln 1 ln R du = 1 p : u p up 1
ln 2
hence 1 n (n + 2) n=1
k X
= =
n=1
1 1 1 1 1 + + + :::: 3 2 4 3 5 1 1 1 1 1 + + + k 2 k k 1 k+1 k 1 1 1 = 1+ + 2 k+1 k+2 1 ! 3 2
k X
1 n
1 n+2
1 k+2
so that
k X 1 1 = lim n (n + 2) k!1 n=1 n (n + 2) n=1
1 X
= lim
k!1
1 1 + k+1 k+2
=
3 2
1
3. (18 pts.) Find the Taylorseries for f (x) = ln x at x = 4. Determine its interval of convergence. Solution: Recall that
1 X n
tn
= = =
1 1 t
;
n=0 1 X
jtj < 1 jtj < 1 jtj < 1
( 1) tn
n=0 1 X ( 1)n tn+1 n+1 n=0
1 ; 1+t
ln (1 + t) ;
let x
4 = t then ln (x) = = ln (4 + t) = ln 4 + ln 1 + ln 4 + t 4 = ln 4 + jx
1 X ( 1)n t n+1 4 ; n+1 n=0
t...

...Fourier series
From Wikipedia, the free encyclopedia
Fourier transforms
Continuous Fourier transform
Fourier series
Discrete-time Fourier transform
Discrete Fourier transform
Fourier analysis
Related transforms
The first four partial sums of the Fourier series for a square wave
In mathematics, a Fourier series (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.
Contents
[hide]
Definition[edit]
In this section, s(x) denotes a function of the real variable x, and s is integrable on an interval [x0, x0 + P], for real numbers x0 and P. We will attempt to represent s in that interval as an infinite sum, or series, of harmonically related sinusoidal functions. Outside the interval, the series is periodic with period P. It follows that if s also has that property, the approximation is valid on the entire real line. The case P = 2π is prominently featured in the literature, presumably because it affords a minor simplification, but at the expense of generality.
For integers N > 0, the following summation is a periodic function with period P:
Using the identities:
Function s(x) (in red) is a sum of six sine functions of different...

...Fourier Series
Fourier series started life as a method to solve problems about the ﬂow of heat through ordinary materials. It has grown so far that if you search our library’s data base for the keyword “Fourier” you will ﬁnd 425 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial diﬀerential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylorseries is based on the idea that you can write a general function as an inﬁnite series of powers. The idea of Fourier series is that you can write a function as an inﬁnite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now. Legendre polynomials are an important example of functions used for such expansions. An example: On the interval 0 < x < L the function x2 varies from 0 to L2 . It can be written as the series of cosines L2 4L2 + 2 x = 3 π
2 ∞ 1
(−1)n nπx cos 2 n L 2πx 1 3πx πx 1 − cos + cos − ··· L 4 L 9 L (1)
=
L2 3
−
4L2 π2
cos
To see...

...[pic] Fourier Series: Basic Results
[pic]
Recall that the mathematical expression
[pic]
is called a Fourier series.
Since this expression deals with convergence, we start by defining a similar expression when the sum is finite.
Definition. A Fourier polynomial is an expression of the form
[pic]
which may rewritten as
[pic]
The constants a0, ai and bi, [pic], are called the coefficients of Fn(x).
The Fourier polynomials are [pic]-periodic functions. Using the trigonometric identities
[pic]
we can easily prove the integral formulas
(1)
for [pic], we have
[pic]
(2)
for m et n, we have
[pic]
(3)
for [pic], we have
[pic]
(4)
for [pic], we have
[pic]
Using the above formulas, we can easily deduce the following result:
Theorem. Let
[pic]
We have
[pic]
This theorem helps associate a Fourier series to any [pic]-periodic function.
Definition. Let f(x) be a [pic]-periodic function which is integrable on [pic]. Set
[pic]
The trigonometric series
[pic]
is called the Fourier series associated to the function f(x). We will use the notation
[pic]
Example. Find the Fourier series of the function
[pic]
Answer. Since f(x) is odd, then an = 0, for [pic]. We turn our attention to the coefficients bn. For any [pic], we have
[pic]
We...

... ELIZABETH TAYLOR
Elizabeth Taylor starred in many movies throughout her lifetime. During her time of her career she married many men that all ended in divorce beside one death. Taylor had two boys and two girls, one of her daughters were adopted. When she got to her last days she helped found an organization for AIDS. She later died but lived a very fulfillment life. Elizabeth Taylor started her life in London, England, yet when she arrived in the United States she made a huge impact in movie star history.
The start of Elizabeth Taylor began with her birth. Elizabeth Rosemund Taylor was born 1932 in London to Francis Taylor and Sara (Corliss). At 10, Elizabeth was fighting for a spot in a program called There’s One Born Every Minute (Corliss). After her first time in the spot light she got her first big break with MGM. MGM producer Sam Marx had a problem: the girl he had cast as the female lead in Lassie Come Home was not right for the part so Elizabeth got the part (Corliss). Though she played in many movies she received only five nominations and one award. Her first Oscar nomination as Best Actress was from the movie Raintree County (Taraborrelli 133). Taylor received her second nomination as playing as a woman with a “voracious” appetite for sex in the movie Cat on a Hot Tin Roof (144). Joseph Mankiewicz had directed Suddenly, Last...

...Taylor Swift
By: Sofia Saenz
Taylor Alison Swift, born December 13, 1989, is an American country pop singer-songwriter, musician and actress.
Taylor Swift was born on December 13, 1989 in Wyomissing, Pennsylvania. She is the daughter of Andrea Gardner, a homemaker, and Scott Kingsley Swift, a stockbroker. Her maternal grandmother, Majorie Finlay, was an opera singer. Taylor Swift has a younger brother, Austin.
At the age of ten, a computer repairman showed her how to play three chords on a guitar, Making Taylor Swift wanting learn the instrument. Afterwards, Taylor Swift wrote her first song, "Lucky You". She began writing songs regularly and used it as an outlet to help her with her pain from not fitting in at school. Taylor Swift was a victim of bullying, and spent her time writing songs to express her emotions. She also started performing at local karaoke contests, festivals, and fairs.
Taylor Swift began to regularly visit Nashville, Tennessee and work with local songwriters. When she was 14, her family relocated to Nashville, Tennessee. Her first major show was a well-received performance at the Bloomsburg Fair. In Tennessee, Taylor Swift attended Hendersonville High School, but was subsequently home schooled for her junior and senior years. In 2008, she earned her high school diploma.
Taylor Swift's greatest musical...