Proof. If , we have
Thus, if the formula is true for
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,
Then, we consider the product
Where in the last line we cancelled terms again. The only things we are left with are the
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
Find the error in an inductive “proof”
Establish a formula for the sản phẩm (1-1/2)(1-1/3)…(1 – 1/n)
January 10, 2021 at 1:09 am
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
March 13, 2019 at 9:51 am
I’m not quite understanding the rearranging step – could you please elaborate?
October 12, 2015 at 1:46 pm
Daniel Fugisawa says:
Amazing! Thank you again!
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
Point out an error, ask a question, offer an alternative solution (to use Latex type
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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.
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