Question: sum cannot return the symbolic summation

We know that sum can return an symbolic summation, for example,


But sometime the relation of parameters in series cannot be defined easily. And it seems that sum cannot correctly determine the symbolic summation of this kind of series:

Maybe it's due to the value of _C(infinite) is undefined. Does anyone have good idea to compute the symbolic summation of this series?

I'd appreciate any help on this topic. Thank a lot.

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