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Find the sum of tens to infinity 1+2/4+2/4.5/8+2/4.5/8.8/12+.......... Infinity

Find the sum of tens to infinity 1+2/4+2/4.5/8+2/4.5/8.8/12+.......... Infinity

Grade:11

2 Answers

Samyak Jain
333 Points
5 years ago
Concept : Binomial expansion of rational index
Let S = 1+2/4+2/4.5/8+2/4.5/8.8/12+.......... \infty
Assume S to be (1 + x)n = 1 + nx + n(n – 1)x2 / 2!+ …..........\infty ,  |x|
So nx = 2/4 = 1/2 ,  n(n – 1)x2 / 2! = 2/4.5/8 = 5/16  or  n(n – 1)x2 = 5/8
n2(n – 1)x2 / n = 5/8  => (n – 1)(nx)2 = 5n/8  => (n – 1)(1/4) = 5n/8
2n – 2 = 5n   =>  n = – 2/3
\therefore x = – 3/4
|x| = 3/4 th term of S.
Therefore, our assumption is correct.
\therefore S = (1 – 3/4)– 2/3 = (1/4) – 2/3  
\therefore S = 42/3
Samyak Jain
333 Points
5 years ago
Concept : Binomial expansion of rational index
Let S = 1+2/4+2/4.5/8+2/4.5/8.8/12+.......... \infty
Assume S to be (1 + x)n = 1 + nx + n(n – 1)x2 / 2!+ …..........\infty ,  |x|
So nx = 2/4 = 1/2 ,  n(n – 1)x2 / 2! = 2/4.5/8 = 5/16  or  n(n – 1)x2 = 5/8
n2(n – 1)x2 / n = 5/8  => (n – 1)(nx)2 = 5n/8  => (n – 1)(1/4) = 5n/8
2n – 2 = 5n   =>  n = – 2/3
\therefore x = – 3/4
|x| = 3/4
Also the values of n & x satisfy 4th term of S.
Therefore, our assumption is correct.
\therefore S = (1 – 3/4)– 2/3 = (1/4) – 2/3  
\therefore S = 42/3
 
 

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