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p-adic numbers and the Ramanujan summation

Linkey
in Mathematics

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I have watched a video about the p-adic numbers:

 

 

Then, there is the Ramanujan's summation which is used in quantum mechanics (Casimir effect):

 

1+2+3+4+5+6...=-1/12

https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension

This forum does not support Latex maths?

Then, we have one more such summation:

1+x+x*x+x*x*x+x*x*x*x+x*x*x*x*x...=-1/(1-x)

This is simple for x<1 and for x>1 we get:

 

1+2+4+8+16+32+64...=-1

1+10+100+1000+10000...=-1/9

1+20+400+8000+160000...=-1/19

 

Can we obtain all these sums with the p-adic numbers?

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