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My response was wrong.

The phenomenon you observe is a special case of a theorem of Frobenius (1880):

If a series is Cesaro summable then it is also Abel summable, and the Cesaro limit is the same as the Abel limit.

In your case the series is $(1,\zeta,\zeta^2,\dots)$ which ensures Cesaro summability as the sequence of partial sums is periodic.

For more details see Page 4 of Korevaar: Tauberian theory - A century of developments (Springer, 2004).

The explanation is simple. The averages of the initial partial sums of a convergent series have the same limit as the partial sums, which is the sum of the series. In particular, if the partial sums form a periodic sequence, then their average over a full period equals the sum of the series.

My response was wrong.

The explanation is simple. The averages of the initial partial sums of a convergent series have the same limit as the partial sums, which is the sum of the series.

The explanation is simple. The averages of the initial partial sums of a convergent series have the same limit as the partial sums, which is the sum of the series. In particular, if the partial sums form a periodic sequence, then their average over a full period equals the sum of the series.