Proof. If , we have

*
on the left, và on the right we have
*
. Thus, the formula is true for .Assume then that the formula is true for some
*
. Then,

*

Thus, if the formula is true for

*
then it is true for
*
. Since we have established that it is true for , we then have that it is true for all
*

Update: From a request in the comments, we’ll địa chỉ in a way lớn arrive at the formula (without just guessing).

First, we write,

*
" title="Rendered by QuickLaTeX.com"/>

Then, we consider the product

*

Where in the last line we cancelled terms again. The only things we are left with are the

*
in the numerator và the 2 &
*
in the denominator. Of course, this is pretty much a proof that the formula is correct without using induction, but it doesn’t rely on us guessing the formula correctly.

As noted in the comments, often it is easier to guess the correct formula và use induction lớn prove the formula is correct than to lớn derive the formula directly.




Bạn đang xem: Basel problem

Related


Share this:


Post navigation
Find the error in an inductive “proof”
Establish a formula for the sản phẩm (1-1/2)(1-1/3)…(1 – 1/n)

10 comments

*

January 10, 2021 at 1:09 am
Anonymous says:

alternatively, you can say that 1-1/4 = 1-(1/2) * 1+(1/2)= 1/2*3/2, và 1-1/9 = 1-(1/3)*(1+(1/3))= 2/3 * 4/3… then 3/2 * 2/3 = 1, & continue like that with the rest of the series, until finally all that is left is 1/2*(1+1/n) = 1/2 + 1/2/n = 1/2(1+1/n)= (1+n)/2n


Reply
*

March 13, 2019 at 9:51 am
Amber says:

I’m not quite understanding the rearranging step – could you please elaborate?


Reply
*

October 12, 2015 at 1:46 pm
Daniel Fugisawa says:

Amazing! Thank you again!


Reply
*

October 12, năm ngoái at 12:46 pm
Daniel Fugisawa says:

Hi! Could you tell me how did you come up with the general law? Thanks


Reply

Point out an error, ask a question, offer an alternative solution (to use Latex type at the top of your comment):Cancel reply


ArchivesArchivesSelect Month June năm 2016 (12) May 2016 (142) April năm nhâm thìn (150) March năm 2016 (156) February 2016 (145) January 2016 (155) December 2015 (155) November 2015 (150) October 2015 (155) September năm ngoái (150) August năm ngoái (155) July 2015 (155) June năm ngoái (50)

If you are having trouble with math proofs a great book lớn learn from is How to lớn Prove It by Daniel Velleman:


*
*

A really awesome book that I highly recommend on how lớn study math và be a math major is Laura Alcock"s, How khổng lồ Study as a Mathematics Major:
*
*

My Complete Math Book Recommendations:
Basics: Calculus, Linear Algebra, và Proof WritingCore Mathematics Subjects.
Send to e-mail AddressYour NameYour email Address
*
Cancel
Post was not sent - kiểm tra your email addresses!
Email kiểm tra failed, please try again


Xem thêm: Quả Và Hạt Do Bộ Phận Nào Của Hoa Tạo Thành, Quả Do Bộ Phận Nào Của Hoa Tạo Thành

Sorry, your blog cannot nói qua posts by email.