We need your funds

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?



>> Challenge Problem 14: Cube Inside Hemisphere Puzzle
>> Challenge Problem 13: What Is The Area of Triangle 1984 AIME?
>> Challenge Problem 12: Tricky Algebra Problem From Nepal Olympiad

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/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. 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!

Source

Mind Your Decision
https://mindyourdecisions.com/blog/2020/01/02/the-answer-is-too-good/

Yêu Tiếng Anh


Loading...

3 comments:

  1. Looking for a good deal on bulk glitter? Explore a wide range of the best bulk glitter on GlitterMall to find one that suits you! Besides good quality brands, you’ll also find plenty of discounts when you shop for bulk glitter during big sales. Don’t forget one crucial step - filter for items that offer bonus perks like free shipping & free return to make the most of your online shopping experience!

    ReplyDelete
  2. colorful compression socks wholesale
    More and more people wear compression socks, there will always be people curious about its principle, want to understand its role, whether it is as magical as the manufacturer said, set a variety of functions in one? Small make up with you to explore the discovery, a comprehensive analysis of compression socks.
    1. Compression socks were first used in the field of treatment
    Compression socks have been used for decades in patients with chronic venous insufficiency to improve venous return and reduce lower extremity swelling and thrombosis by applying a top-down gradient of pressure.
    As we all know, the heart drives the blood flow throughout the body. When the blood flows back to the veins through the arteries and capillaries, the power has been significantly attenuated, and the lower extremity venous blood needs to overcome gravity to do work to return to the heart. Will the venous blood flow back? Hey hey, the human body is magic, vein blood vessels usually have the existence of double valve, in the flow from the bottom to the top, the valve is attached to the vascular wall, will not cause obstruction, once standing or other reasons caused by venous pressure increase, two valves will open, prevent venous blood reflux.

    ReplyDelete

  3. blood pressure watch professional in R&D wearable blood pressure monitor, in the innovative form of a wrist watch, proactively monitors your heart health by turning real-time heart data into heart.
    mason jar lids canning lids supplier. offer canning Lids in different style and size
    Cervical spine massager We aim to create a healthy lifestyle that assists every individual to de-stress, relax. unwind and streamline a pain-free life and at the same time feel and look amazing.
    sex machine for her Choose Sex Machine to Get Orgasm and Enjoy Life
    moving dildo Enjoy sexual life for both men and women
    neon light for roomNeon Signs Light Is One-of-a-Kind Activities Designed and Hosted by Expert Locals. All Experiences are Vetted for Quality.
    dog training collarsPets Supplier mall offers the ultimate pet shop experience. We have all the pet supplies, pet food, toys and accessories you and your pet needs at great prices. Find all the best pets suppliers coupons, promotion.
    novelty giftsDiscover the latest and most interesting gadgets at a bargain price
    Car decorationHere you will find the latest and most fashionable car accessories

    ReplyDelete

♥ You are highly recommended to use Google or Facebook accounts to jot a few comments down if you have any further questions or concerns. Don't forget to check on "Notify me" at the bottom-right corner of the comment box to keep tracking your comment replies.
♥ Share to be shared - if you want to share any of your own thoughts to this blog's readers, please don't hesitate to reach me here. Many Thanks!
Have a good one!

Related Posts Plugin for WordPress, Blogger...