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March 22, 2020

Challenge Problem 15: An Amazing Result of Complicated Exponent Problem

This puzzle was created by a MindYourDecisions fan in India. What is the value of the following infinite product?

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The numerators are the odd nth roots of e and the denominators are even nth roots of e.

First, write each term as a fractional exponent and then use exponent rules to simplify.

= (e^1. e^-1/2)(e^1/3. e^-1/4)(e^1/5. e-1/6)…
= e^1. e^-1/2.e^1/3. e^-1/4.e^1/5. e-1/6…
= e^(1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + …)

The exponent is the alternating harmonic series. What is its value? Recall the series:

ln(1 + x) = x – x^2/2 + x^3/3 + … (for -1 < x ≤ 1)

Substitute x = 1 to get:

ln(1 + 1) = 1 – 1/2 + 1/3 + …
ln(2) = 1 – 1/2 + 1/3 + …

The sum of the alternating harmonic series is ln(2). Thus the answer to the original problem is:

e^ln(2) = 2

The answer is super amazing!


Mind Your Decision

Yêu Tiếng Anh



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