So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$
s.t. at every non-negative integer, k $$f(x)=\sum_{n=0}^{\infty}{a_n(x-k)}$$
locally and the radius of convergence of this power series is exactly one. I constructed what seems to me to be theoretically an interesting function, and I'm not sure if this was the conventional answer to the problem (though it probably is). I constructed the function: $$f(x):=\sum_{k=0}^{\infty}{\frac {1}{x^2-2kx+k^2+1}\cdot \frac {1} {2^k}}= \sum_{k=0}^{\infty}{\frac {1}{(x-k)^2+1}\cdot \frac {1} {2^k}}$$
The first thing you can notice is that computationally it may be better in some circumstances to use a nicer scaling factor than the 1/2^k in order to calculate numerical approximations since this function approaches $0$ very quickly away from the origin. Mathematica seems to support my intuition that this function has poles at $k+i$ and $k-i$ for all non-negative integers $k$. (I recommend people play around with this function numerically or analytically in Mathematica as its quite fun) And also, we can conclude it must since it converges uniformly by the M-test to an analytic function on $\mathbb R$ (since each partial sum is analytic, at least on compact subsets not containing the poles). (Is this correct reasoning?)
Can we conclude that this function actually converges everywhere to a meromorphic function on $\mathbb C$?
Finally, my last question on this topic is if there is a way to tackle this problem assuming less or no complex analysis knowledge at all? This was a problem in my real analysis class and I feel like we weren't expected to know this complex information about radii of convergence. (I don't think he taught us this)
Please let me know of any errors in my post that I should correct. Any other thinking on this topic is happily welcomed as well. Thanks. :)
NOTE: According to Mathematica it seems you can construct this same sum without scaling each subsequent rational function down. Why does this converge and how can we be sure? And analytically Mathematica says this function is related to the PolyGamma function. I am very interested in this now... hopefully somebody can give me more reference on this topic.