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Sum with alternate signs
02-06-2015, 02:26 PM (This post was last modified: 02-06-2015 02:28 PM by salvomic.)
Post: #1
Sum with alternate signs
hi,
there is a way in Prime to do this sum?
\[ \sum_{k=1}^{\infty}{\frac {(-1)^{k+1}}{k^{2}} } \]

the value is \( \frac {π^{2}}{12} \)

HP Prime gives symbolic form, not the value of the sum...

Thanks

Salvo

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02-06-2015, 04:47 PM
Post: #2
RE: Sum with alternate signs
Same problem in xcas and Maxima. Wolframalpha gives the right answer.
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02-06-2015, 04:49 PM
Post: #3
RE: Sum with alternate signs
(02-06-2015 04:47 PM)retoa Wrote:  Same problem in xcas and Maxima. Wolframalpha gives the right answer.

yes, in fact!
As I like much more Prime (and HP 50g), I wonder why they don't...

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02-06-2015, 05:08 PM (This post was last modified: 02-06-2015 05:17 PM by Gilles.)
Post: #4
RE: Sum with alternate signs
(02-06-2015 02:26 PM)salvomic Wrote:  hi,
there is a way in Prime to do this sum?
\[ \sum_{k=1}^{\infty}{\frac {(-1)^{k+1}}{k^{2}} } \]

the value is \( \frac {π^{2}}{12} \)

HP Prime gives symbolic form, not the value of the sum...

Thanks

Salvo

You can do

\[ \sum_{k=1}^{\infty}{\frac {-1}{(2*k)^{2}} } + \sum_{k=1}^{\infty}{\frac {1}{(2*k-1)^{2}} } \]

By the way I get the correct answer on the HP50G but my Prime seems unable to calculate Psi(1/2,1) in a numeric value.

I get :

1/4*Psi(1/2,1)-Pi²/24

Same on 50G then ->NUM returns 0.8224...
On the Prime ~ don't 'solve' Psi(0.5,1) . Strange ...
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02-06-2015, 05:10 PM (This post was last modified: 02-06-2015 05:17 PM by retoa.)
Post: #5
RE: Sum with alternate signs
I also tried to decompose it in

\( \sum_{k=1}^{\infty}(\frac{1}{(2k-1)^2}-\frac{1}{(2k)^2}) \)

to avoid the (-1)^(k+1), but I did not get the wanted result. Still the Psi(1/2,1)
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02-06-2015, 05:18 PM
Post: #6
RE: Sum with alternate signs
(02-06-2015 05:08 PM)Gilles Wrote:  You can do

\[ \sum_{k=1}^{\infty}{\frac {-1}{(2*k)^{2}} } + \sum_{k=1}^{\infty}{\frac {1}{(2*k-1)^{2}} } \]

By the way I get the correct answer on the HP50G but my Prime seems unable to calculate Psi(1/2,1) in a numeric value.

thanks a lot, Gilles,
yes I see that Prime don't approx Psi1/2,1); my HP50 does it.

Hope in a next firmware to have the symbolic result (π^2/12), more interesting than Psi() Smile

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02-06-2015, 06:55 PM
Post: #7
RE: Sum with alternate signs
Indeed, for the approx value of Psi(x,1), Xcas calls the GSL, that is not available on the Prime.
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02-06-2015, 07:03 PM
Post: #8
RE: Sum with alternate signs
(02-06-2015 06:55 PM)parisse Wrote:  Indeed, for the approx value of Psi(x,1), Xcas calls the GSL, that is not available on the Prime.

I understand.
There is no other way to approximate Psi on Prime?

thank you

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02-07-2015, 06:46 AM
Post: #9
RE: Sum with alternate signs
No built-in yet. Maybe I'll implement something, in the meantime you can write a user program
http://people.math.sfu.ca/~cbm/aands/page_260.htm
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02-07-2015, 10:11 AM
Post: #10
RE: Sum with alternate signs
(02-07-2015 06:46 AM)parisse Wrote:  No built-in yet. Maybe I'll implement something, in the meantime you can write a user program
http://people.math.sfu.ca/~cbm/aands/page_260.htm

ok, thank you for information!
I'll think to write a program, maybe... Smile

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05-13-2015, 08:09 PM
Post: #11
RE: Sum with alternate signs
the problem is now solved with the firmware 7820!

Answer: -π/12

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