Determine if there is an analytic function in |z|<2 to take, in the z = 1 / n, n = 1,2,3, ..., the values ​

Determine if there is an analytic function in |z|<2 to take, in the z = 1 / n, n = 1,2,3, ..., the values ​​given:

a)1,2/3,3/5,4/7,5/9,6/11,7/13,8/15,...

b)1/2.-1/2,1/3,-1/3,1/4,-1/4,1/5,-1/5,…

Thanks for your help

a) We have f(1/n) = n/(2n-1) = 1/(2 - 1/n) for all n = 1, 2, ...

Since {1/n} converges to 0 which is in |z| < 2, we conclude that
f(z) = 1/(2 - z) by the Identity Theorem. (Check: f is analytic in |z| < 2.)
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b) Note that f(1/(2n-1)) = 1/(n+1) = 2/((2n-1) + 3).
As in (a), we see that f(z) = 2/(2z + 3); however this is not analytic at z = -3/2 which is in |z| < 2.

So, f does not exist.

(Alternately, f(1/(2n)) = -1/(n+1) = -2/((2n-1) + 3) ==> f(z) = -2/(2z+3), and
we can't have 2/(2z + 3) = -2/(2z + 3)...)

I hope this helps!