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Converges vs. diverges

ptc-4307506
1-Newbie

Converges vs. diverges

Why can mathcad not show my whether this series converges or diverges?

9 REPLIES 9
RichardJ
19-Tanzanite
(To:ptc-4307506)

Your expression makes no sense. n is defined in the summation as ranging from 1 to infinity, so what do you expect the limit to do?

Is this what you actually wanted to do?

Richard Jackson wrote:

Your expression makes no sense. n is defined in the summation as ranging from 1 to infinity, so what do you expect the limit to do?

Maybe Rasmus had something like this in mind

series.png

RichardJ
19-Tanzanite
(To:Werner_E)

Yes, that would make sense

The limes you wrote in front does not make sense, as the result of the series would not include any variable n!

The series itself (without the lim) diverges, but Mathcads symbolics is not powerful enough to come to that conclusion - it would require comparisons with well know similar diverging series, which Mathcad obviously has not built in.

The limes (without the series) converges against 0, but of course you won't need Mathcad to see that.

So there is no way Mathcad can tell my whether the series diverges or converges?

RichardJ
19-Tanzanite
(To:ptc-4307506)

Not directly, no. You can use it to apply standard tests though. See for example

http://math2.org/math/expansion/tests.htm

and

http://www.math.hawaii.edu/~ralph/Classes/242/SeriesConvTests.pdf

Rasmus Jørgensen wrote:

So there is no way Mathcad can tell my whether the series diverges or converges?

At least until now nobody here has found a way to do so directly..

Whether matchad can show it or not, the series diverges. The nth term is always greater than (1/2)*n^(-1/2), which diverges. The latter term n^(-p), without the factor 1/2, is the general term of a series for the Riemann zeta function, which diverges for p <=1, and converges for p>1.

Mathcad symbolics usually need a helping hand, as has been pointed out many times. In mcd11, the symbolics got two different answers for the referecne series - one correct, one not - with a seemingly innocuous change.

Lou

Werner_E
24-Ruby V
(To:LouP)

Its a shame that newer Mathcad versions with Mupad as symbolic engine have still more problems with that rather simple limes. Only the float-mode (how should we call that mode which is automatically taken if MC symbolics encounters even a single decimal point - "semi-exact"?) it returns the right answer.

diverge1.png

That "float-trick" however does not work for the original limit:

diverge2.png

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