Date: Oct 6, 2017 12:05 PM
Author: Mike Terry
Subject: Re: Can two series, both diverges, multiplied give a series that<br> converges?
On 06/10/2017 16:21, konyberg wrote:

> fredag 6. oktober 2017 17.13.31 UTC+2 skrev Peter Percival fĂ¸lgende:

>> konyberg wrote:

>>> Consider these two series. s = lim (n=1 to inf) Sum(1/n) and t = lim

>>> (n=1 to inf) Sum(1/(1+n)). Both series diverges, going to infinity.

>>> Now if we multiply these,

>>

>> What is the definition of the product of two infinite series?

>>

>>

>>> we can argue that every product of the new

>>> series is smaller or equal to 1/n^2. So it should converge. Or can

>>> we? Let us write the first as a series without the sigma and the

>>> other with sigma. s*t = (1+1/2+1/3+ ...) * t. And since the first

>>> from s (1 * t) diverges, how can s*t converge?

>>>

>>> KON

>>>

> It is the multiplication of the two series.

That doesn't answer Peter's question. Each series has infinitely many

terms, and you need to say what you mean the product to be calculated

from those terms.

If you thought this through carefully, you'd realise straight away the

answer to your original question, I think!

To get you started in the right direction, suppose the first series is:

Sum [n=1 to oo] (a_n)

and the second is:

Sum [n=1 to oo] (b_n)

Now, what do you mean by the "product" of these series?

If you feel tempted to reply "just multiply them together", then ask

yourself "multiply WHAT together exactly?" (Remember, multiplication is

an operation that takes TWO numbers, and gives a single number as the

answer. In the two series, you have INFINITELY many numbers...)

Or perhaps your answer will be that the product of the two series is

some new third series? (If so, then say what is the n'th term of this

new series?)

Regards,

Mike.