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I have an expression to define some coordinates in a region between $X_l$ and $X_u$, where $X_u > X_l$:

$$X_n = X_l + dx \sum_{n=0}^{N} (1-\delta _{n,0}) A^n $$

For $n = 0$ this gives $X_0 = X_l$. My aim is to find the number of terms, $N$, required for $X_N = X_u$ assuming all other terms in the expression are known.

The summation over the power series is where I'm struggling - remove the summation and this is simply rearrange and use a log to find n, but I'm not sure how to manipulate the expression with the summation present. I feel like this has a simple solution I'm overlooking, but any thoughts are appreciated.

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