symbolic - How to symbolically differentiate an infinite series without evaluating the series itself

I'm dealing with finite sums of infinite series. Each of the infinite series possesses a different starting index, i.e. each of the series begins at n = 0, n = 1, or n = 2. As a result, it's important to keep track of the indices for each of the series.

In addition, Mathematica cannot evaluate the series, because it's too complicated and undesirable for what's to be accomplished. I've only written the series in this manner to allow for efficient symbolic manipulation.

The problem is that Mathematica takes a long time "trying" to evaluate each of the series. I've tried Hold, HoldAll, Unevaluated, and HoldForm. HoldForm seems like the correct choice, but there's a problem. Although I don't want to evaluate the series themselves, I'd still like to be able to perform operations on the series, such as differentiation.

So, does anyone know of a way to differentiate a series without evaluating the series itself?

I'm new to Mathematica, so the answer might be simple. Thanks for your time.

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