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Infinite Series Convergence, how?

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Infinite Series Convergence, how?

Hello,

I have attached my worksheet.

Is there a way for mathcad to tell me if an infinite series converges or diverges? And if it converges, what it converges to?

Thank you!!

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Re: Infinite Series Convergence, how?

You can solve for a specific accuracy

View solution in original post

22 REPLIES 22

Re: Infinite Series Convergence, how?

As ratio and root tests fail you have to use comparison to prove that this series converges (which does not mean that you now know what it converes to).

To evaluate that kind of series with Mathcad you have to use symbolic evaluation, not the numeric one you tried

14.04.png

Unfortunately Mathcads symbolics isn't capable to deal with your series. The results are different whether you force float mode (using 0.5) or exact mode (using 1/2), but not helpful in either case.

14.04.png

Mathcad is generally not really good with that kind of symbolics. While it knows about this series and successfully yields the Rieman Zeta as result

14.04.png

it isn't even able to deal with this (equivalent) series

14.04.png

Re: Infinite Series Convergence, how?

Thank you for the help.

Can Matlab do this?

Re: Infinite Series Convergence, how?

Can Matlab do this?

Matlab uses the very same symbolics processor (muPad, they own it), so I would guess - no. But if you have access, just give it a try.

Re: Infinite Series Convergence, how?

Would it matter if I were able to tell MathCAD to compute the value within a specific accuracy?

Re: Infinite Series Convergence, how?

david h wrote:

Would it matter if I were able to tell MathCAD to compute the value within a specific accuracy?

No. Limiting precision can't help. You have an infinite series, so summing up "all" summand would take an infinite time. That means that Mathcad would have to find some kind of closed formula to be able to compute it and as it seems is not able to find one. BTW, if you write 0.5 Mathcads symbolics automatically(unfortunately) switch to float mode with limited precison (you could force it and control precision using the "float" modifier) and yields "undefined" as you have seen.

The numeric evaluation on the other hand is (for obvious reasons) not able to sum up to infinity and so fails if you put infinity as upper limit.

Best chances (if any) with Mathematica, maybe also Maple. You may also give the Wolfram Alpha webpage a try, who knows?

Re: Infinite Series Convergence, how?

The Maple engine in earlier versions of Mathcad can't deal with it either.

Also, this is what Wolfram Alpha comes up with:

series.png

So I'm not sure Mathematica would give you an answer either.

Re: Infinite Series Convergence, how?

Good to know. I guess there will be no easy closed form of the limit. Have no acceess to Mathematica at the time here.

Re: Infinite Series Convergence, how?

Can't insert pictures when editing the post above. 😞

I never tried this before, but it looks like Mathcad can at least tell us that the series is bounded above (lower bound 0 is obvious anyway); nte that this does not necessarily mean its convergent.

At least it works when we write 0.5, not with 1/2

15.04.png

But there are two things I don't understand:

15.04.png

I always thought that using a decimal point in symbolic evaluation forces an automatic float, but obviously there is a difference

Re: Infinite Series Convergence, how?

I always thought that using a decimal point in symbolic evaluation forces an automatic float, but obviously there is a difference

That was also my understanding. So we just learned something, even if it's not what the sum converges to

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