Maclaurin's theorem and expansion.
Answers
Answer:
Maclaurin's Theorem
This simple article shows how the Maclaurin series works, and how to write out the expansions. Although modern calculators are able to show the expansions, it is worth learning them for basic functions such as sin x and cos x.
The expansion above shows the general formula of the Maclaurin series. I have animated it to make it simpler for students to understand. The series is one of the easiest to understand in mathematics and very repetitive as you can imagine. If you are ever reading a scientific paper, you will find mathematicians making a meal of this one to make their work appear more complicated than it really is. I always have a chuckle when I see one of these dressed up.
I was just recently reading some of the original manuscripts of Madhava, and Srinivasa Ramanujan and the amazing thing I noticed was that they did not have fancy names for their formulas. They were simply looking for mathematical steps that would work every time. Moreover, they were relying heavily on intuition and foresight, which is actually very refreshing to see because that is the thinking process required to make progress in this field. Therefore, in this article I will avoid all the confusing jargon and get straight into the interesting stuff that works. Once you have understood the principle, you can call it whatever you like, get bogged down by terminology, and limits.
The Expansion
The series describes the steps required to convert any function f(x) into its equivalent series expansion. Many functions can be expressed as a series expansion, but not all. How will you know if a function has a series equivalent? If the function has derivatives then there is a good chance it may expand as a series. As long as the numerator part is not zero, it has a good chance, however, the only way is to have a go and find out. Usually, trigonometry functions such as sine, cosine, tangent, and natural log Ln may expand as a series. This is how digital calculators are able to give you the sine or cosine values of any angle.
All you have to do is to find the values of the bits in red and plug them into the series. The red bit are called the derivatives of the function and you simply find the derivative and set x to zero to find its value.
Step-by-step explanation:
A Maclaurin series is a Taylor series expansion of a function about 0,
f(x)=f(0)+f^'(0)x+(f^('')(0))/(2!)x^2+(f^((3))(0))/(3!)x^3+...+(f^((n))(0))/(n!)x^n+....
(1)
Maclaurin series are named after the Scottish mathematician Colin Maclaurin.
The Maclaurin series of a function f(x) up to order n may be found using Series[f, {x, 0, n}]. The nth term of a Maclaurin series of a function f can be computed in the Wolfram Language using SeriesCoefficient[f, {x, 0, n}] and is given by the inverse Z-transform
a_n=Z^(-1)[1/x](n).
Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series.
Maclaurin series for common functions include
1/(1-x)=1+x+x^2+x^3+x^4+x^5+...
for -1<x<1
fn(x,k)=1-1/2x^2+1/(24)(1+4k^2)x^4+...
cosx=1-1/2x^2+1/(24)x^4-1/(720)x^6+...
for -infty<x<infty /(112)x^7-...
(8)
for -1<x<1
(9)
coshx=1+1/2x^2+1/(24)x^4+1/(720)x^6+1/(40,320)x^8+...
(10)
cot^(-1)x=1/2pi-x+1/3x^3-1/5x^5+1/7x^7-1/9x^9+...
(11)
dn(x,k)=1-1/2k^2x^2+1/(24)k^2(4+k^2)x^4+...
(12)
erf(x)=1/(sqrt(pi))(2x-2/3x^3+1/5x^5-1/(21)x^7+...)
(13)
e^x=1+x+1/2x^2+1/6x^3+1/(24)x^4+...
(14)
for -infty<x<infty
(15)
_2F_1(alpha,beta;gamma;x)=1+(alphabeta)/(1!gamma)x+(alpha(alpha+1)beta(beta+1))/(2!gamma(gamma+1))x^2+...
(16)
ln(1+x)=x-1/2x^2+1/3x^3-1/4x^4+...
(17)
for -1<x<=1
(18)
ln((1+x)/(1-x))=2x+2/3x^3+2/5x^5+2/7x^7+...
(19)
for -1<x<1
(20)
secx=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+(277)/(8064)x^8+...
(21)
sechx=1-1/2x^2+5/(24)x^4-(61)/(720)x^6+(277)/(8064)x^8+...
(22)
sinx=x-1/6x^3+1/(120)x^5-1/(5040)x^7+...
(23)
for -infty<x<infty
(24)
sin^(-1)x=x+1/6x^3+3/(40)x^5+5/(112)x^7+(35)/(1152)x^9+...
(25)
sinhx=x+1/6x^3+1/(120)x^5+1/(5040)x^7+1/(362880)x^9+...
(26)
sinh^(-1)x=x-1/6x^3+3/(40)x^5-5/(112)x^7+(35)/(1152)x^9-...
(27)
sn(x,k)=x-1/6(1+k^2)x^3+1/(120)(1+14k^2+k^4)x^5+...
(28)
tanx=x+1/3x^3+2/(15)x^5+(17)/(315)x^7+(62)/(2835)x^9+...
(29)
tan^(-1)x=x-1/3x^3+1/5x^5-1/7x^7+...
(30)
for -1<x<1
(31)
tanhx=x-1/3x^3+2/(15)x^5-(17)/(315)x^7+(62)/(2835)x^9+...
(32)
tanh^(-1)x=x+1/3x^3+1/5x^5+1/7x^7+1/9x^9+....
(33)
The explicit forms for some of these are
1/(1-x)=sum_(n=0)^(infty)x^n
(34)
cosx=sum_(n=0)^(infty)((-1)^n)/((2n)!)x^(2n)
(35)
cos^(-1)x=pi/2-sum_(n=0)^(infty)(Gamma(n+1/2))/(sqrt(pi)(2n+1)n!)x^(2n+1)
(36)
coshx=sum_(n=0)^(infty)1/((2n)!)x^(2n)
(37)
cot^(-1)x=pi/2-sum_(n=0)^(infty)((-1)^n)/(2n+1)x^(2n+1)
(38)
e^x=sum_(n=0)^(infty)1/(n!)x^n
(39)
erf(x)=sum_(n=0)^(infty)(2(-1)^n)/(sqrt(pi)(2n+1)n!)x^(2n+1)
(40)
_2F_1(alpha,beta;gamma,x)=sum_(n=0)^(infty)((alpha)_n(beta)_n)/((gamma)_n)(x^n)/(n!)
(41)
ln(1+x)=sum_(n=1)^(infty)((-1)^(n+1))/nx^n
(42)
ln((1+x)/(1-x))=sum_(n=1)^(infty)2/((2n-1))x^(2n-1)
(43)
secx=sum_(n=0)^(infty)((-1)^nE_(2n))/((2n)!)x^(2n)
(44)
sechx=sum_(n=0)^(infty)(E_(2n))/((2n)!)x^(2n)
(45)
sinx=sum_(n=0)^(infty)((-1)^n)/((2n+1)!)x^(2n+1)
(46)
sin^(-1)x=sum_(n=0)^(infty)(Gamma(n+1/2))/(sqrt(pi)(2n+1)n!)x^(2n+1)
(47)
sinhx=sum_(n=0)^(infty)1/((2n+1)!)x^(2n+1)
(48)
sinh^(-1)x=sum_(n=0)^(infty)(P_(2n)(0))/(2n+1)x^(2n+1)
(49)
tanx=sum_(n=0)^(infty)((-1)^n2^(2n+2)(2^(2n+2)-1)B_(2n+2))/((2n+2)!)x^(2n+1)
(50)
tan^(-1)x=sum_(n=1)^(infty)((-1)^(n+1))/(2n-1)x^(2n-1)
(51)
tanhx=sum_(n=1)^(infty)(2^(2n)(2^(2n)-1)B_(2n))/((2n)!)x^(2n-1)
(52)
tanh^(-1)x=sum_(n=1)^(infty)1/(2n-1)x^(2n-1),
(53)
where Gamma(x) is a gamma function, B_n is a Bernoulli number, E_n is an Euler number and P_n(x) is a Legendre polynomial.