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>>> If it's a limit, just compute it for numbers as close as possible to
>>> the limiting value (or really really big numbers if the limiting
>>> value is infinite). The closer (or the bigger) the number you
>>> compute it for the closer your answer will be.
>>
>> Wouldn't that be increadibly unstable, numerically?
>
> How much of a problem this is depends on the limit.
True enough.
>>>> But you can't compute an infinite product.
>>>
>>> If the product converges then you can by definition get an
>>> arbitrarily good approximation by computing the product of the first
>>> n terms for a large enough n.
>>
>> I don't see how that is the case.
>>
>> If you have an infinite *sum*, then as long as the terms get
>> progressively more tiny and never get larger again, you can disregard
>> all the terms after a certain point. But if you're taking a *product*
>> then any term, anywhere in the series could radically alter the final
>> result.
>
> The same is true for a sum actually. The point is proving that an
> infinite sum or product *converges* is to show that this doesn't happen.
> Basically you show that the partial sums/products form a convergent
> sequence (see the "Limit of a sequence" Wikipedia article I linked).
> Then you know by definition that you can get a good approximation by
> computing a large enough partial sum/product.
>
> In the case of a sum this requires that the terms progressively get
> closer to zero, for a product it'll require they get closer to one.
I guess it's just the case that an infinite sum or product can be
convergent, and yet converge really, *really* slowly...
>>> Out of curiosity have you ever had a calculus class?
>>
>> I've never had *any* maths class!
>>
>> (Unless you count what we did at school. This simply involved filling
>> out hundreds of thousands of pages of long-division problems over a
>> 7-year period...)
>>
>> Hypothetically I shouldn't be able to do algebra at all...
>
> That's too bad, it seems like you would have greatly enjoyed being able
> to do some math that was more advanced than long division.
Well, I *did* go to a school for stupid people, after all...
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