What is the sum?

[Steven G]

Let A be the set of all positive integers whose factors are 2, 3 and 5.

If you add the reciprocals of all the numbers of A, then what will the sum be?

1/1 + 1/2 + 1/3 +1/4 +1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + 1/16 + 1/18 + 1/20 + ...

 

[Steven G]

I don't understand why 1/1 is there and 1/14 is not there...

The numbers in A are all in the form 2^a*3^b*5^c.

So 1 = 2⁰*3⁰*5⁰, while 14 can not be written in the form 2^a*3^b*5^c

Do you understand????

 

[Deleted member 4993]

The numbers in A are all in the form 2^a*3^b*5^c.

So 1 = 2⁰*3⁰*5⁰, while 14 can not be written in the form 2^a*3^b*5^c

Do you understand????

Good eye Jomo! I had myself confused too!

Denis trained you well - now you can beat him in these "numbers" game!!!

 

[Dr.Peterson]

Let A be the set of all positive integers whose factors are 2, 3 and 5.

If you add the reciprocals of all the numbers of A, then what will the sum be?

1/1 + 1/2 + 1/3 +1/4 +1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + 1/16 + 1/18 + 1/20 + ...

(1/1 + 1/2 + 1/4 + ...)(1/1 + 1/3 + 1/9 + ...)(1 + 1/5 + 1/25 + 1/125 + ...) = 1/(1-1/2) * 1/(1-1/3) * 1/(1 - 1/5) = 2 * 3/2 * 5/4 = 3.75

Each term in the expansion of the product is a product 1/2^m * 1/3^n * 1/5^p.

 

[Dr.Peterson]

I don't understand why 1/1 is there and 1/14 is not there...

Let A be the set of all positive integers whose factors are 2, 3 and 5.

But, to be precise, none of 2, 3, and 5 are factors of 1. And 2, 3, and 5 are not all factors of many of the denominators. So the problem statement could be improved. Perhaps something like

Let A be the set of all positive integers with no prime factors except 2, 3 and 5. ...
​

 

[Steven G]

Denis trained you well - now you can beat him in these "numbers" game!!!

But Denis cheats by using a computer! I bet he even uses a calculator as well. It is Denis who needs to be trained! The problem is that Denis is not trainable.

 

[HallsofIvy]

But Denis cheats by using a computer!

And you count on your fingers?

(Every time I see "using technology" I wonder if that includes paper and pencil. They, after all, require technology to be made.)

I bet he even uses a calculator as well. It is Denis who needs to be trained! The problem is that Denis is not trainable.