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Topic:
Can two series, both diverges, multiplied give a series that converges?
Replies:
22
Last Post:
Oct 7, 2017 12:52 AM
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Re: Can two series, both diverges, multiplied give a series that
converges?
Posted:
Oct 6, 2017 12:05 PM
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On 06/10/2017 16:21, konyberg wrote:
> fredag 6. oktober 2017 17.13.31 UTC+2 skrev Peter Percival følgende:
>> 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.
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