tommy's "linear" summability method

[tommy1729]

tommy's summability method
**************************

I might not be the first to consider it so the name is perhaps disputable or temporarily anyway ,


This method is for divergeant strictly increasing positive integer sums

consider the geometric sum : 1 + 2 + 4 + 8 + ... + 2^n + ... = x

now we see 2 x = 2 + 4 + 8 + ...

notice x - 1 = 2 + 4 + 8 + ...

so

2 x = x - 1.

thus

x = -1.


lets generalize for real a > 1 :

1 + a + a^2 + ... = x

then

a x = x - 1

so

x = -1/(a-1) = 1/(1-a).

This is all old stuff.

So whats next ?

consider an entire function f(x) given by a taylor series : f_1 x + f_2 x^2 + f_3 x^3 + ... + f_i x^i + ...

such that f(i+1) > f(i) for every integer i > 0.

Notice f(0) = 0.


Now we can sum f(1) + f(a) + f(a^2) + f(a^3) + ... + f(a^n) + ... = x

Here is how

sum f(1) + f(a) + f(a^2) + f(a^3) + ... + f(a^n) + ... = x

sum_n sum_i  f_i (a^n)^i = x

so

x = sum_n sum_i  f_i (a^n)^i = sum_i sum_n  f_i (a^i)^n

= sum_i f_i * - 1/(a^i - 1) = sum_i f_i / (1 - a^i)



As example consider

dexp(x) := exp(x) - 1.

dexp(1) + dexp(2) + dexp(2^2) + dexp(2^3) + dexp(2^4) + ... = x

Then

x  = sum_i f_i /( 1 - 2^i ) = sum_i  1/( i! * (1 - 2^i) )


this clearly converges.

(I got an estimate of x = -1.19355 btw)

I call this a linear summability method because it has some linear properties.



regards


tommy1729

[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.

---

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

[jacob]

1 + 3 + 9.... = -0.5