MacLaurin series

This graph show the use of a MacLaurin series to approximate the sine of x, or sin(x), when x = 0, and other values determined by polynomials with degree 1, 3, 5, 7, 9, 11 and 13. Credit: IkamusumeFan.

A MacLaurin series is a Taylor series that has a term at (0,0).

Calculus

This diagram shows an approximation to an area under a curve. Credit: Dubhe.

Notation: let the symbol $\Delta$ represent change in.

Notation: let the symbol $d$ represent an infinitesimal change in.

Notation: let the symbol $\partial$ represent an infinitesimal change in one of more than one.

Let

y = f(x)

be a function where values of $x$ may be any real number and values resulting in $y$ are also any real number.

\Delta x

is a small finite change in

x

which when put into the function

f(x)

produces a

\Delta y

These small changes can be manipulated with the operations of arithmetic: addition ( $+$ ), subtraction ( $-$ ), multiplication ( $*$ ), and division ( $/$ ).

\Delta y = f(x + \Delta x) - f(x)

Dividing $\Delta y$ by $\Delta x$ and taking the limit as $\Delta x$ → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

f(x) = x^2

f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2

\Delta y/\Delta x = (x^2 + 2x\Delta x + (\Delta x)^2 - x^2)/\Delta x

\Delta y/\Delta x = 2x + \Delta x

as $\Delta x$ and $\Delta y$ go towards zero,

dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.

This ratio is called the derivative.

Let

y = f(x,z)

then

\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z

\partial y/ \partial x = \partial f(x,z)

where z is held constant and

\partial y / \partial z = \partial f(x,z)

where x is held contstant.

Notation: let the symbol $\nabla$ be the gradient, i.e., derivatives for multivariable functions.

\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.

For

\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.

Any particular individual rectangle for a sum of rectangular areas is

f(x_i + \Delta x_i) * \Delta x_i.

The approximate area under the curve is the sum $\sum$ of all the individual (i) areas from i = 0 to as many as the area needed (n):

\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.

Notation: let the symbol $\int$ represent the integral.

limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.

This can be within a finite interval [a,b]

\int_a^b f(x) \; dx

when i = 0 the integral is evaluated at $a$ and i = n the integral is evaluated at $b$ . Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

Def. a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a calculus.

"Calculus [focuses] on limits]], functions, derivatives, integrals, and infinite series."^[1]

"Although calculus (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero)."^[2]

Series

Taylor Series:

$y= \sum_{n=0}^{\infty }\frac{f^{n}(a)(x-a)^{n}}{n!}=f(a)+{f}'(a)(x-a)+ \frac{{f}''(a)(x-a)^{2})}{2!}+\frac{{f}'''(a)(x-a)^{3}}{3!}+...,$

where fⁿ refers to the number (n) of derivatives taken.

A MacLaurin series of a function ƒ(x) for which a derivative may be taken of the function or any of its derivatives at 0 is the power series

f(0)+\frac {f'(0)}{1!} (x)+ \frac{f''(0)}{2!} (x)^2+\frac{f^{(3)}(0)}{3!}(x)^3+ \cdots.

which can be written in the more compact sigma, or summation, notation as

\sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}

where n! denotes the factorial of n and ƒ⁽ⁿ⁾(0) denotes the nth derivative of ƒ evaluated at the point 0. The derivative of order zero ƒ is defined to be ƒ itself and (x)⁰ and 0! are both defined to be 1.

MacLaurin series for e^x

Taylor series is defined as

\sum_{n=0} ^ {\infin } \frac {f^{(n)}(t)}{n!} \, (x-t)^{n}

$\displaystyle$

The MacLaurin series occurs when t=0

\sum_{n=0} ^ {\infin } \frac {f^{(n)}(0)}{n!} \, (x)^{n}

$\displaystyle$

The derivatives are

{y}'=e^x

{y}''=e^x

{y}'''=e^x

Development of MacLaurin series for $e^x$

e^x=\frac{1}{0!}+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3...

$\displaystyle$

e^x=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3...

Explicit form can be written as

e^x=\sum_{n=0} ^ {\infin } \frac {1}{n!} \, (x)^{n}

MacLaurin series for sin(x)

y=sin(t)

y'=cos(t)

y''=-sin(t)

y'''=-cos(t)

y^{iv}=sin(t)

y^{v}=cos(t)

Development of MacLaurin series for $sin(x)$

sin(x)=\frac{0}{0!}+\frac{1}{1!}x-\frac{0}{2!}x^2-\frac{1}{3!}x^3+\frac{0}{4!}x^4+\frac{1}{5!}x^5...

$\displaystyle$

sin(x)=1x-\frac{1}{6}x^3+\frac{1}{120}x^5...

Explicit form can be written as

sin(x)=\sum_{n=0} ^ {\infin } \frac {-1}{(2n+1)!} \, (x)^{(2n+1)}

MacLaurin series for cos(x)

Development of MacLaurin series for $cos(x)$

y=cos(t)

y'=-sin(t)

y''=-cos(t)

y'''=sin(t)

y^{iv}=cos(t)

cos(x)=\frac{1}{0!}+\frac{0}{1!}x-\frac{1}{2!}x^2+\frac{0}{3!}x^3+\frac{1}{4!}x^4...

$\displaystyle$

cos(x)=1-\frac{1}{2}x^2+\frac{1}{24}x^4...

Explicit form can be written as

cos(x)=\sum_{n=0} ^ {\infin } \frac {-1}{2n!} \, (x)^{2n}

Euler's formula

Recalling Euler's Formula:

e^{i\omega x}=\cos\omega x+i\sin\omega x\!

Recall the Taylor Series from above for $e^x$ at : $t=0$ (also called the MacLaurin series)

e^{x}=\sum_{n=0}^{\infty}\frac{x^n}{n!}\!

By replacing x with $i\omega x$ , the Taylor series for $e^{i\omega x}$ can be found:

e^{i\omega x}=\sum_{n=0}^{\infty}\frac{(i\omega x)^n}{n!}\!

even powers of n = 2k:

i^{2k}=(i^2)^k=(-1)^k\!

odd powers of n = 2k+1:

i^{2k+1}=(i^2)^k i=(-1)^k i\!

For $i\omega x$ :

e^{i\omega x}=\sum_{n=0}^{\infty}\frac{i^n (\omega x)^n}{n!}=\sum_{k=0}^{\infty}\frac{i^{2k} (\omega x)^{2k}}{(2k)!}+\sum_{k=0}^{\infty}\frac{i^{2k+1} (\omega x)^{2k+1}}{(2k+1)!}\!

Using the two previous equations:

e^{i \omega x}=\sum_{k=0}^{\infty}\frac{(-1)^{k} (\omega x)^{2k}}{(2k)!}+i\sum_{k=0}^{\infty}\frac{(-1)^{k} (\omega x)^{2k+1}}{(2k+1)!}\!

\Rightarrow e^{i \omega x}=\cos (\omega x)+i \sin (\omega x)\!

Therefore, the first part of the equation is equal to the Taylor series for cosine, and the second part is equal to the Taylor series for sine as follows:

\cos( \omega x)=\sum_{k=0}^{\infty}\frac{(-1)^{k} ( \omega x)^{2k}}{(2k)!}\!

\sin ( \omega x)= \sum_{k=0}^{\infty}\frac{(-1)^{k} ( \omega x)^{2k+1}}{(2k+1)!}\!

MacLaurin series for $\displaystyle (1-x)^{-a}$

$\displaystyle f(x) = \frac {1}{(1-x)^a}$

Table for Maclaurin Series
$f(x) = \frac {1}{(1-x)^a}$	$f(0) = \frac {1}{(1-0)^a}=1$
$f '(x) = \frac {a}{(1-x)^{a+1}}$	$f '(0) = \frac {a}{(1-0)^{a+1}} = a$
$f' '(x) = \frac {a(a+1)}{(1-x)^{a+2}}$	$f' '(0) = \frac {a(a+1)}{(1-0)^{a+2}} = a(a+1)$
$f^{(3)}(x) = \frac {a(a+1)(a+2)}{(1-x)^{a+3}}$	$f^{(3)}(0) = \frac {a(a+1)(a+2)}{(1-0)^{a+3}}= a(a+1)(a+2)$
And so on..	..

Rewriting the Maclaurin series expansion,

\displaystyle \frac {1}{(1-x)^a} = f(0)+\frac {f'(0)}{1!} (x-0)+ \frac{f''(0)}{2!} (x-0)^2+\frac{f^{(3)}(0)}{3!}(x-0)^3+ \cdots.

Substituting the values from the table, we get

\displaystyle \frac {1}{(1-x)^a} = 1 +\frac {a}{1!} (x)+ \frac{a(a+1)}{2!} (x)^2+\frac{a(a+1)(a+2)}{3!}(x)^3+ \cdots.

\displaystyle \frac {1}{(1-x)^a} = \sum_{n=0} ^ {\infin } (a)_k \frac {x^k}{n!}

Using

(a)_0 := 1

(a)_k := a(a+1)(a+2)\cdots(a+k-1)

We can represent

\sum_{n=0} ^ {\infin } \frac{(a)_k \, (b)_k}{(c)_k}\, \frac {x^k}{n!} = F(a,b;c;x)

\sum_{n=0} ^ {\infin } \frac{(a)_k (b)_k}{(b)_k}\frac {x^k}{n!} = F(a,b;b;x)

MacLaurin series for $\displaystyle arctan(1+x)$

We have the function

$\displaystyle arctan(1+x)$

Expand $\displaystyle arctan(1+x)$

Table for Maclaurin Series
$f(x) = arctan(1+x)$	$f(0) = arctan(1+0) = \frac{\pi}{4}$
$f '(x) = \frac {1}{1+(1+x)^{2}}$	$f '(0) = \frac {1}{1+(1+0)^{2}} = \frac {1}{2}$
$f' '(x) = \frac {-2(x+1)}{\left[1+(1+x)^2\right]^2}$	$f' '(0) = \frac {-2(0+1)}{\left[1+(1+0)^2\right]^2}= \frac{-1}{2}$
$f^{(3)}(x) = \frac {2(3x^2+6x+2)}{\left[1+(1+x)^2\right]^3}$	$f^{(3)}(0) = \frac {2(3(0)^2+6(0)+2)}{\left[1+(1+0)^2\right]^3} = \frac{1}{2}$
And so on..	..

Rewriting the Maclaurin series expansion,

\displaystyle arctan(1+x) = f(0)+\frac {f'(0)}{1!} (x-0)+ \frac{f''(0)}{2!} (x-0)^2+\frac{f^{(3)}(0)}{3!}(x-0)^3+ \cdots.

Substituting the values from the table, we get

\displaystyle arctan(1+x) = \frac{\pi}{4} + \frac {1}{2*1!} (x)- \frac{1}{2*2!} (x)^2+\frac{1}{2*3!}(x)^3+ \cdots.

MacLaurin series for $arctan(x)$

$arctan(x)$

Expanding $arctan(x)$ using Maclaurin's series

Table for Maclaurin Series
$f(x) = arctan(x)$	$f(0) = arctan(0) = 0$
$f '(x) = \frac {1}{1+x^{2}}$	$f '(0) = \frac {1}{1+(0)^{2}} = 1$
$f' '(x) = \frac {-2x}{\left[1+(x)^2\right]^2}$	$f' '(0) = \frac {-2(0)}{\left[1+(0)^2\right]^2}= 0$
$f^{(3)}(x) = \frac {6x^2-2}{\left[1+(x)^2\right]^3}$	$f^{(3)}(0) = \frac {6(0)^2-2)}{\left[1+(0)^2\right]^3} = -2$
$f^{(4)}(x) = \frac {-24x(x^2-1}{\left[1+(x)^2\right]^4}$	$f^{(4)}(0) = \frac {-24(0)((0)^2-1}{\left[1+(0)^2\right]^4}= 0$
$f^{(5)}(x) = \frac {24(5x^4-10x^2+1}{\left[1+(x)^2\right]^5}$	$f^{(5)}(x) = \frac {24(5(0)^4-10(0)^2+1}{\left[1+(0)^2\right]^5} = 24$
And so on..	..

Rewriting the Maclaurin series expansion,

\displaystyle arctan(x) = f(0)+\frac {f'(0)}{1!} (x-0)+ \frac{f''(0)}{2!} (x-0)^2+\frac{f^{(3)}(0)}{3!}(x-0)^3+ \cdots.

Substituting the values from the table, we get

\displaystyle arctan(x) = 0 +\frac {1}{1!} (x)+ \frac{0}{2!} (x)^2+\frac{-2}{3!}(x)^3+ \frac{0}{4!}(x)^4 + \frac{24}{5!}(x)^5 \cdots.

Engineering

The "performance of a Markov system under different operating strategies [can be estimated] by observing the behavior of the system under the [strategy of having] a Maclaurin series for the performance measures of [the] Markov chains."^[3]

Research

Hypothesis:

Any non-convergent function can be represented by a MacLaurin series.

Control groups

This is an image of a Lewis rat. Credit: Charles River Laboratories.

The findings demonstrate a statistically systematic change from the status quo or the control group.

“In the design of experiments, treatments [or special properties or characteristics] are applied to [or observed in] experimental units in the treatment group(s).^[4] In comparative experiments, members of the complementary group, the control group, receive either no treatment or a standard treatment.^[5]"^[6]

Proof of concept

Def. a “short and/or incomplete realization of a certain method or idea to demonstrate its feasibility"^[7] is called a proof of concept.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of

background,
procedures,
findings, and
interpretation.^[8]

References

↑ "Calculus, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. October 13, 2012. Retrieved 2012-10-14.
↑ "infinitesimal calculus, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Setember 19, 2012. Retrieved 2013-01-31.
↑ Xi-Ren Cao (1998). "The Maclaurin Series for Performance Functions of Markov Chains". Advances in Applied Probability 30: 676-92. http://www.ece.ust.hk/~eecao/paper/60.pdf. Retrieved 2014-07-23.
↑ Klaus Hinkelmann, Oscar Kempthorne (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (2nd ed.). Wiley. ISBN 978-0-471-72756-9. http://books.google.com/?id=T3wWj2kVYZgC&printsec=frontcover.
↑ R. A. Bailey (2008). Design of comparative experiments. Cambridge University Press. ISBN 978-0-521-68357-9. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521683579.
↑ "Treatment and control groups, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. May 18, 2012. Retrieved 2012-05-31.
↑ "proof of concept, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. November 10, 2012. Retrieved 2013-01-13.
↑ Ginger Lehrman and Ian B Hogue, Sarah Palmer, Cheryl Jennings, Celsa A Spina, Ann Wiegand, Alan L Landay, Robert W Coombs, Douglas D Richman, John W Mellors, John M Coffin, Ronald J Bosch, David M Margolis (August 13, 2005). "Depletion of latent HIV-1 infection in vivo: a proof-of-concept study". Lancet 366 (9485): 549-55. doi:10.1016/S0140-6736(05)67098-5. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1894952/. Retrieved 2012-05-09.

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MacLaurin series

Calculus

Series

MacLaurin series for ex

MacLaurin series for sin(x)

MacLaurin series for cos(x)

Euler's formula

MacLaurin series for

MacLaurin series for

MacLaurin series for

Engineering

Research

Control groups

Proof of concept

See also

References

External links

MacLaurin series for e^x

MacLaurin series for $\displaystyle (1-x)^{-a}$

MacLaurin series for $\displaystyle arctan(1+x)$

MacLaurin series for $arctan(x)$