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gatlas11-Garnet
11-Garnet
April 23, 2023
Solved

Unable to expand an expression into a series because of a singularity.

  • April 23, 2023
  • 2 replies
  • 2243 views

Mathcad won't allow me to expand the attached expression into a series because of a singularity. Yet the publication shows that it can be done.

 

Any suggestions? Please see the attachment.

Best answer by Werner_E

"series" expands an expression into a Taylor series. If not otherwise stated at u=0, which in your case of course must fail because of the division by zero if u=0.

A Taylor series is a (most often inifinte) polynomial, a sum of powers of u (or of u-u0). The series you want to see sure is not a polynomial and so you can't expect "series" to give you that answer.

Furthermore "series" never returns an infinite sum but only finite ones (which usually means that you get just an approximation of the initial expression).

You could tell Prime to give you a Taylor series at a position different from u=0, maybe u=1

Werner_E_0-1682249892232.png

but thats not what you would like to see.

 

I see no way to let Prime expand the expression to that sum shown in your pic.

You could hope that a symbolic Math program may be able to do it the other way round, simplifying the series to the initial expression, but I guess that Primes symbolic is not capable enough to do so.

Werner_E_1-1682249960386.png

 

Wolfram Alpha e.g. is able to get rid of the sums and so succeeds in simplification, even though the result is not exactly the one you are starting with

Werner_E_2-1682250186236.png

 

BTW, I don't think that the sum in the picture and your initial expression are equivalent for negative values of u.

 

 

2 replies

T
21-Topaz II
terryhendicott21-Topaz II
21-Topaz II
April 23, 2023

Hi,

Series and original equation are close 8% to 1% but not equal.  For symbolics they have to be equal.

Capture.JPG

Cheers

Terry

G
gatlas11-GarnetAuthor
11-Garnet
April 23, 2023

Terry and Ruby,

 

I was able to generate a single series using a substitution. However, I can't make it into a "simple summation."

 

Please see the attachment.

 

Regards,

Glen

W
Werner_E25-Diamond I
25-Diamond I
April 23, 2023
@gatlas wrote:

Terry and Ruby,

 

I was able to generate a single series using a substitution. However, I can't make it into a "simple summation."

 

Please see the attachment.

 

Regards,

Glen

With "simple summation" you mean an infinite sum, nicely displayed with the sum operator? As I already wrote in my first answer, Prime will not return that kind of display, not even for the Taylor series. It will only return a finite sum of powers, a finite polynomial. Its up to you to find out the formation law for the coefficients. Of course you can use normal calculus to do and Prime may help with the derivatives.

BTW, you may do the substitution and series in one go

Werner_E_0-1682260636571.png

Werner_E_1-1682260649211.png

and you may also follow it by the re-substitution

Werner_E_2-1682260726757.png

or

Werner_E_3-1682260744322.png

 

You can create completely different sums using different substitutions

Werner_E_6-1682261424004.png

 

or

Werner_E_5-1682261316675.png

 

but what for? What is the goal?

 

I was surprised to read that you seemed to consider the sum you achieved as being a partial success.

But this sum has nothing to do with the sum shown in the picture you posted. Its a completely different sum, even if you would be able to find its generic formula and could write it using the sum symbol.

So I wonder what the ultimate goal of your efforts should actually be?

 

BTW, Ruby isn't my name, its part of a silly ranking scheme here where names of stones are used. Additionally the rank expressed as a number (looks like I rank 24, whatever that may mean), and additionally we seem to have some kind of sub-ranks, at least I think thats what the "V" after "Ruby" should be. And because all of this nonsense is not enough, you also see a rank-speedometer beside the name. They could never really explain to me why this whole bunch of nonsense would be necessary.
And that's not all for the kids, there are also special icons, badges you earn, etc.
But don't let that irritate you, it's still a forum by and for adults, even if it doesn't look like it on first glance because all of this 😉

W
Werner_E25-Diamond IAnswer
25-Diamond I
April 23, 2023

"series" expands an expression into a Taylor series. If not otherwise stated at u=0, which in your case of course must fail because of the division by zero if u=0.

A Taylor series is a (most often inifinte) polynomial, a sum of powers of u (or of u-u0). The series you want to see sure is not a polynomial and so you can't expect "series" to give you that answer.

Furthermore "series" never returns an infinite sum but only finite ones (which usually means that you get just an approximation of the initial expression).

You could tell Prime to give you a Taylor series at a position different from u=0, maybe u=1

Werner_E_0-1682249892232.png

but thats not what you would like to see.

 

I see no way to let Prime expand the expression to that sum shown in your pic.

You could hope that a symbolic Math program may be able to do it the other way round, simplifying the series to the initial expression, but I guess that Primes symbolic is not capable enough to do so.

Werner_E_1-1682249960386.png

 

Wolfram Alpha e.g. is able to get rid of the sums and so succeeds in simplification, even though the result is not exactly the one you are starting with

Werner_E_2-1682250186236.png

 

BTW, I don't think that the sum in the picture and your initial expression are equivalent for negative values of u.

 

 

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