Find the sum of the series 1+(1/3)+(1.3/3.6)+(1.3.5/3.6.9)+.......... is
Find the sum of the series 1+(1/3)+(1.3/3.6)+(1.3.5/3.6.9)+.......... is
MERIN VARGHESE , 6 Years ago
Grade
Algebra> Find the sum of the series 1+(1/3)+(1.3/3...
1 AnswersArun
Last Activity: 6 Years ago
This is simply a particular case of the binomial series (follow the link below to read an article):
(1 + x)^n = ∑[k=0 .. ∞] n(n-1)(n-2) . . (n-k+1) x^k / k!,
having n = -1/2 and x = -2/3:
(1 - 2/3)^(-1/2) = 1 + (-1/2) * (-2/3) + (-1/2)(-3/2) * (-2/3)^2/2! +
+ (-1/2)(-3/2)(-5/2) * (-2/3)^3/3! + (-1/2)(-3/2)(-5/2)(-7/2) * (-2/3)^4/4! + . . =
= 1 + 1/3 + (1*3/2^2)(2^2/3^2)/2! + (1*3*5/2^3)(2^3/3^3)/3! + (1*3*5*7/2^4)(2^4/3^4)/4! + . . . =
/cancel the degrees of 2 now/
= 1 + 1/3 + 1*3/(3^2*2!) + 1*3*5/(3^3*3!) + 1*3*5*7/(3^4*4!) + . . =
= 1 + 1/3 + 1*3/(3*6) + 1*3*5/(3*6*9) + 1*3*5*7/(3*6*9*12) + . . . = (1/3)^(-1/2) = √3
(1 + x)^n = ∑[k=0 .. ∞] n(n-1)(n-2) . . (n-k+1) x^k / k!,
having n = -1/2 and x = -2/3:
(1 - 2/3)^(-1/2) = 1 + (-1/2) * (-2/3) + (-1/2)(-3/2) * (-2/3)^2/2! +
+ (-1/2)(-3/2)(-5/2) * (-2/3)^3/3! + (-1/2)(-3/2)(-5/2)(-7/2) * (-2/3)^4/4! + . . =
= 1 + 1/3 + (1*3/2^2)(2^2/3^2)/2! + (1*3*5/2^3)(2^3/3^3)/3! + (1*3*5*7/2^4)(2^4/3^4)/4! + . . . =
/cancel the degrees of 2 now/
= 1 + 1/3 + 1*3/(3^2*2!) + 1*3*5/(3^3*3!) + 1*3*5*7/(3^4*4!) + . . =
= 1 + 1/3 + 1*3/(3*6) + 1*3*5/(3*6*9) + 1*3*5*7/(3*6*9*12) + . . . = (1/3)^(-1/2) = √3
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Other Related Questions on algebra
Can we divide two vectors? If this is possible then how?
Can we divide two vectors? If this is possible then how?
algebra
1 Answer Available
Last Activity: 1 Year ago
How do you write “the product of 18 and q” as an algebraic expression?
How do you write “the product of 18 and q” as an algebraic expression?
algebra
1 Answer Available
Last Activity: 1 Year ago
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे
algebra
0 Answer Available
Last Activity: 2 Years ago