Mathematics Dictionary Dr. K. G. Shih

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Binomial Theory

Subjects

Symbol Defintion Example : x^2 = square of x

AN 16 00 | - Outline AN 16 01 | - Binomial Theory AN 16 02 | - Expand 1/(1+x) into theory AN 16 03 | - Expand 1/(1-x) into theory AN 16 04 | - Expand 1/(1+x^2) into theory AN 16 05 | - Expand 1/(1-x^2) into theory AN 16 06 | - Prove that Lim[(1 + 1/x)^x] = e AN 16 07 | - expand Sqr(1 + x^2) AN 16 08 | - AN 16 09 | - AN 16 10 | - AN 16 00 | - Outines

Answers

AN 16 01. Binomial theory Definition (x + y)^n = x^n + C(n,1)*(x^(n-1))*(y^1) + C(n,2)*(x^(n-2))*(y^2) + .... C(n,r) is binomial coefficient C(n,r) = (n*(n-1)*(n-2)*....(n-r+1)/(r!) Examples of C(n,r) : Symmetrical pattern C(n,0) = 1 C(n,1) = n C(n,2) = n*(n-1)/2! C(n,n-2) = n*(n-1)*(n-2)*...(3)*(2)/(n-1)! = n*(n-1)/2! C(n,n-1) = n C(n,n) = 1 Reference Study Subjects | Algebra : Binomial theory Go to Begin AN 16 02. Expand 1/(1 + x) into series Expand using binomial theory 1/(1 + x) = (1 + x)^(-1) = 1 + (-1)*x + (-1)(-1-1)*x^2/2! + (-1)*(-1-1)*(-1-2)*x^3/3! + ...... = 1 - x + x^2 - x^3 + x^4 - x^5 + .... This can be used to express special functions into series Go to Begin AN 16 03. AN 16 03. expand 1/(1 - x) into series Expand using binomial theory 1/(1 - x) = (1 - x)^(-1) = 1 + (-1)*(-x) + (-1)(-1-1)*(-x)^2/2! + (-1)*(-1-1)*(-1-2)*(-x)^3/3! + ...... = 1 + x + x^2 + x^3 + x^4 + x^5 + .... This can be used to express special functions into series Go to Begin AN 16 04. expand 1/(1 + x^2) into series Expand using binomial theory 1/(1 + x^2) = (1 + x^2)^(-1) = 1 + (-1)*(x^2) + (-1)(-1-1)*(x^2)^2/2! + (-1)*(-1-1)*(-1-2)*(x^2)^3/3! + ...... = 1 - x^2 + x^4 - x^6 + x^8 - x^10 + .... This can be used to express special functions into series Go to Begin AN 16 05. expand 1/(1 - x^2) into series Expand using binomial theory 1/(1 - x^2) = (1 + x^2)^(-1) = 1 + (-1)*(-x^2) + (-1)(-1-1)*(-x^2)^2/2! + (-1)*(-1-1)*(-1-2)*(-x^2)^3/3! + ...... = 1 + x^2 + x^4 + x^6 + x^8 + x^10 + .... Note This can be used to express special functions into series Go to Begin AN 16 06. Prove that Lim[(1 + 1/x)^x] = e as x goes to infinite Graphic solution Subjects Limit Prove using binomial theory Binomial theory (1 + 1/x)^x = 1 + x*(1/x) + (x*(x-1)*(1/x)^2)/2! + (x*(x-1)(x-2)*(1/x)^3)/3! + ..... Lim[(x*(x-1)*(1/x)^2)/2!] = Lim[(1*(1-1/x)]/2! = 1/2! as x goes to infintit Lim[(x*(x-1)(x-2)*(1/x)^3)/3!] = Lim[(1*(1-1/x)*(1-2/x)]/3! = 1/3! as x goes to infintit Hence Lim[(1 + 1/x)^2] = 1 + 1 + 1/2! + 1/3! + ..... The series of e^x e^x = 1 + x + (x^2)/2! + (x^3)/3! + ..... Let x = 1 Hence e = 1 + 1 + 1/2! + 1/3! + ..... Hence Lim[(1 + 1/x)^2] = e as x goes to infinite Change to a new limit expression Let 1/x = u, u goes to zero as x goes to infinite Lim[(1 + u)^(1/u)] = e as u goes to zero This limit is used to find the derivative of ln(u) Go to Begin AN 16 07. Expand Sqr(1 + x^2) into series See Subjects AL 07 12 Go to Begin AN 16 08. Answer Go to Begin AN 16 09. Answer Go to Begin AN 16 10. Answer Go to Begin AN 16 00. Outlines Express function as series 01. Binomial theory 02. 1/(1 + x) = 1 - x + x^2 - x^3 + ... 03. 1/(1 - x) = 1 - x + x^2 - x^3 + ... 04. 1/(1 + x^2) = 1 - x^2 + x^4 - x^6 + ... 05. 1/(1 - x^2) = 1 + x^2 + x^4 + x^6 + ... 06. Lim[(1 + x)^(1/x)] = e if x goes to infinite 07. Lim[(1 + 1/x)^(x)] = e if x goes to 0 08. e^x = 1 + x + (x^2)/2! + (x^3)/3! + ..... 09. e = 1 + 1 + 1/(2!) + 1/(3!) + ..... 10. Sqr(1 + x^2) = Application in calculus 10. Study Subjects | Calculus : Series in CA 18 00 Go to Begin

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