tommy's "linear" summability method

[tommy1729]

I want to point out some problems :

1) SUMMABILITY METHODS WORK GREAT TO GET A VALUE OF AN ANALYTIC CONTINUATION BUT OFTEN FAIL TO DETECT ( TRUE ) DIVERGEANCE !!

we still want to know about divergeance , also beyond the analytic continuation !

2) the entire function is an interpolation 

so f(2^n) is defined as an f that grows like say f(x) = exp(x).

but almost any entire f that agrees with exp(x) at integer x can be used ... but might give different results !!

So a very disturbing thing.

3)

the sum 1 + 2 + 4 + ...

is associated to the equation 2 x = x - 1

or 4 x = x - 3
8 x = x - 7 

etc

ALL of them give x = -1.

HOWEVER

the sum 1 + 3 + 9 + ...

has equations

3 x = x - 1

9 x = x - 1 - 3 = x - 4

etc

all with DIFFERENT solutions.

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these " problems " are quite universal with summability methods.

Which is why " the master forbids it " ( being Weierstrass )

I do not forbid it , but be very careful.


regards

tommy1729