# Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $$\text{ais}()$$ already here)

Motivation
I'm considering functions, represented by Carleman matrices of infinite size, for instance $$f(x)=t^x - 1$$. Let us denote the iterates of $$f(x)$$ by indexes on $$x$$ like $$x_0=x$$, $$x_1=f(x)$$, $$x_2 = f(f(x))$$ and also $$x_{-1} = \log_t(1+x)$$ and so on. For $$t$$ in the interval $$1 < t < e$$ and $$x_0$$ in an interval from zero to some small limiting value we have convergence for $$x_{n}$$ and $$x_{-n}$$ for $$n \to \infty$$.

Then I look at the infinite alternating series (" ais ": "alternating iteration series") $$\text{ais}(x)= \cdots \pm x_{-k} \cdots - x_{-3} + x_{-2} - x_{-1} + x_0 - x_1 + x_2 - x_3 \cdots \pm x_k \cdots \tag 1$$ and for practical reasons (numerical computations using Pari/GP and its "sumalt()"-function) I define $$\begin{array} {rl} \text{ais}_n(x_0)&= x_0 - x_{-1} + x_{-2} - x_{-3} + \cdots \pm x_{k} \cdots \\ \text{ais}_p(x_0)&= x_0 - x_1 + x_2 - x_3 + \cdots \pm x_k \cdots \\ \text{ais}(x_0) &= \text{ais}_n(x_0) + \text{ais}_p(x_0) - x_0 \end{array} \tag 2$$ Because $$x_{-n}$$ converge to a nonzero value one can also use Cesaro- or Euler-summation for the alternating series $$\text{ais}_n(x)$$ .

The matrix-ansatz...
To save computation time I thought I can make use of the Neumann-series of the Carleman-matrices $$T$$ for $$f(x)=t^x-1$$ and $$T^{-1}$$ for $$f°^{-1}(x)=\log(1+x)/\log(t)$$ such that with a vector $$V(x)=[1,x,x^2,x^3,...]$$ I can write
$$V(x) \cdot T = V(t^x-1) \\ V(x) \cdot T^{-1} = V(\log_t(1+x)) \tag 3$$ and using the Neumann-series make a shortcut to $$V(x_0) \cdot ( I - T + T^2 - T^3 + \cdots - \cdots )=V(x_0) - V(x_1)+ V(x_2) - \cdots + \cdots$$
(All this is computationally simple to check and to manually approximate because $$T$$ and $$T^{-1}$$ are triangular and have also the Carleman-structure.)

So using $$A_p = (I + T)^{-1}$$ and its second column only I get, as expected (...drumroll...) $$V(x_0) \cdot A_p[,1] = \text{ais}_p(x_0) \tag 4$$

...and the conceptional problem
However, this is not so with the supposed Neumann-series of its inverse $$A_n = (I + T^{-1})^{-1}$$. Here I get , with some error-term $$d(x_0)$$, $$V(x_0) \cdot A_n[,1] = \text{ais}_n(x_0) + d(x_0) \tag 5$$

Unfortunately the analogy to the geometric series of scalars breaks here: for the geometric series of a scalar argument we can analytically continue $$a_p(x) = 1-x+x^2-x^3 + \cdots - \cdots = { 1 \over 1 + x} \\ a_n(x)=1-x^{-1}+x^{-2}-x^{-3} + \cdots - \cdots = { 1 \over 1 + x^{-1}} \\ a_p(x) + a_n(x) - x^0 = 0 \tag 6$$ for any $$x \ne \{-1,0\}$$

Now the Neumann-series is not defined if the powers of $$T^{-1}$$ are not converging. However, because the occuring deviation $$d(x_0)$$ looks small and somehow systematic (much sinusoidal) I hope there might be some correction-factor to make sense to the general ansatz $$A_n = (I + T^{-1})^{-1}$$. Perhaps something like the integral-term for the Ramanujan-summation or some extension to the matrix $$A_n$$.
I've searched for articles on the Neumann-series, but didn't find any which deals with the divergent case. So my question:

Q: is something known whether there is some systematic correction-factor for the correction $$d(x_0)$$ in the case where the Neumann-series diverges? The Neumann-series for $$T^{-1}$$ does not converge for this class of examples.

Remark: I have already a solution which allows to compute correctly $$\text{ais}(x)$$ with the Carleman-matrix-method: using two different such matrices. Here I'm asking specifically for some analytical access to the problem of divergence in the Neumann-series.

Update: A plot of $$d(x)$$ for $$x=\{x_{-5} \ldots x_5\}$$ for some base $$t$$ which I use for numerical tests shows nearly perfect sinusoidal behave with pseudo-period $$2h$$ where $$h$$ is the iteration-height. Of course the heights $$h$$ are not linear with the $$x$$ coordinate, so we see that decreasing wavelength at the right and left border. That frappant sinusoidal behave inspired my hope that the correction-factor $$d(x)$$ might be analytically expressible at all:

   *(Pari/GP)* ploth(x=itx(x0,5),itx(x0,-5),maisx(x,5))
*%1052 = [0.0466657282692, 4.24284898742, -0.000432707353441, 0.000432710734738]*

• If $T^{-1}$ and $(I+T)^{-1}$ exist, $(I+ T^{-1})^{-1} = T (I+T)^{-1}$ and this is $\sum_{j=1}^\infty (-1)^{j+1} T^j$ if the latter series converges. – Robert Israel Oct 18 '18 at 15:41
• @RobertIsrael - hmm, if we have $(I+T)^{-1}$ and $T(I+T)^{-1}$ then this is together $(I+T)^{-1} +T(I+T)^{-1} = (I+T)(I+T)^{-1} = I$ - as long as we can assume existence. But that does not work for the determination of the $\text{ais}()$ - we would get $V(x_0) \cdot I[,1] = x_0$ but crosschecked by the serial summation we get the nonzero difference $d(x_0)$ . Or did I misunderstand your comment? – Gottfried Helms Oct 18 '18 at 17:08