The horrible summation

Probability Level 4

1 + 1 4 + 1 ⋅ 3 4 ⋅ 8 + 1 ⋅ 3 ⋅ 5 4 ⋅ 8 ⋅ 12 + ⋯ \large 1+ \dfrac{1}{4}\ +\dfrac{1\cdot 3}{4\cdot8}\ + \dfrac{1\cdot3\cdot 5}{4\cdot8\cdot12}+\cdots

Find the value of the closed form of the sum above. Give your answer to 3 decimal places.


The answer is 1.4142.

Vijay Bhava by Vijay Bhava , 21, India

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2 solutions

Shubham Rustagi
Jan 10, 2017

Think the expansion of binomial

(1 + x)^n = 1 + nx + {(n)(n-1)(x^2)}÷2! + And so on Compare the terms of this expansion with question expansion.

To get n= -0.5 and x= -0.5 Put value in binomial expansion to get ans as √2 🎊🎊👏

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It was really a good solution

Vijay Bhava - 4 years, 5 months ago
Ashutosh Sharma
Jan 27, 2018

Use expansion (1+x)^n=1+nx+n(n-1)/2 x^2+... Rest is not needed. Now compare, so nx=1/4 and n(n-1)/2 x^2 =3/24. Square the nx term and divide and thus find n. So n comes out as -0.5 and so is x. Thus ans is sqrt2= 1.414

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