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Infinite Series Convergence, how?

SOLVED

Re: Infinite Series Convergence, how?

Why don't you say from th every beginning you want to know what the sum converges to :

15.04.png

Will I ever understand that program (especially the symbolics)?

Re: Infinite Series Convergence, how?

???????

Weird!

Re: Infinite Series Convergence, how?

Richard Jackson wrote:

???????

Weird!

Indeed! Symbolics sometimes is a miracle.

Re: Infinite Series Convergence, how?

Quite!

water+wine.png

Re: Infinite Series Convergence, how?

I see you use the biblical Mathcad version 😉

15.04.png

Re: Infinite Series Convergence, how?

You can solve for a specific accuracy

View solution in original post

Re: Infinite Series Convergence, how?

Thank you for everybody's help! You are very gracious for taking your time to help me. Hopefully someday I can help others as well.

Re: Infinite Series Convergence, how?

I believe that Fred's error criterion is incorrect, in that it stops the summation once the nth term is less than the input "error." However, there are still an infinite number of remaining terms, and the terms do not decrease rapidly enough to give a reasonable error bound based on the last term summed.(The last term does bound the error when the terms have decreasing magnitudes and the signs alternate - not the case here.)

The series converges, as can be easily shown. However, the actual error of a partial sum from the infinite summation is significantly larger than the magnitude of the last term summed.

The attached file has an approximation to the infinite sum within an arbitrary and calculable error bound (with the limits of the numeric calculation). I see the key to these types of problems as trying to manipulate the series so that only finite sums are needed in the approximations and the error bounds.

I find that the symbolic processor is not useful in leading the way to finding numeric bounds, as in this case, but it is useful to check the algebraic manipulations I wind up doing by hand.

Lou

P.S. I spent a lot more time on this than I would ever admit to, but I found the problem intriguing.

Re: Infinite Series Convergence, how?

Wow, impressive! Good job. You really spent some with that series 😉

And Mathcads symbolic agrees with your result:

17.04.png

Re: Infinite Series Convergence, how?

I didn't think that the symbolic processor would calc a numeric result for an infinite sum. In MCD11 symbolics returns only the same original sum regardelss of the float value; doesn't attempt an evaluation. I tried the symbolic sum as you show in MCD15 and in Prime2, with your result in both cases. However, if the float limit is 20 or higher, then the symbolic answer is "undefined," in both versions, leading me to think that "19" did not appear in your evaluation by coincidence. Any idea why the boundary exists or what may cause it?

Lou

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