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Sum of reciprocals ?

lvl107
20-Turquoise

Sum of reciprocals ?

Hi, Everyone.
From the following:

I.png

 

How to find the sum of riciprocal, the above ?


Thanks in advance for your time and help.
Best Regards.
Loi.

1 ACCEPTED SOLUTION

Accepted Solutions
ttokoro
20-Turquoise
(To:lvl107)

image.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.png

The sum seems to be 1.

View solution in original post

2 REPLIES 2
ttokoro
20-Turquoise
(To:lvl107)

image.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.pngimage.png

The sum seems to be 1.

Werner_E
24-Ruby V
(To:lvl107)

Mathcads symbolics seems not to be able to cope with this sum and thats no surprise.

Werner_E_0-1664800751634.png

You can rewrite your sum using the Catalan identity, but to no avail

Werner_E_0-1664814787857.png

 

 

For special calculations in number theory you should not use Mathcad but rather a specialized CAS.
A widespread one is PARI/GP which can be downloaded here: http://pari.math.u-bordeaux.fr

 

P.S.: I haven't tried if PARI can come up with the desired result 1, though.

EDIT: I tried using the online browser version of PARI/GP (https://pari.math.u-bordeaux.fr/gp.html), but the result rather looks like a numeric one, not a symbolic 1

Werner_E_1-1664815856278.png

So PARI sure can handle very large numbers and has handy and valuable function when dealing with number theory, but it can't be used to symbolically prove that the infinite sum is exactly 1.

 

 

P.P.S.: I noticed that the numerator and denominator of the (fully reduced) fractions in my examples always differ by exactly 1.
Is this a coincidence? 🤔

Werner_E_0-1664801540544.png

 

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