I'm trying to figure out how the series $\sum g_k$ where $g_k(x) = 0$ if $x \leq k$ and $g_k(x) = (-1)^k$ if $x>k$ converges (pointwise, uniformly or not at all) but I'm having trouble proving what I think. I think that the series converges pointwise but not uniformly, but I'm having trouble because I'm not sure what the limit that the series converges to is, and where to go from there.
2026-03-27 17:51:48.1774633908
Convergence of piecewise function
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Assuming the sum starts at $k = 0$. For a hint, note that the series converges to zero when $x \leq 0$, since each term $g_k(x)$ in the series is just zero. For $0 < x \leq 1$, $g_0(x) = (-1)^0 = 1$, and for all $k \geq 1$, $g_k(x) = 0$. So for $0 < x \leq 1$, the sum evaluates to 1. Try to continue this for successive intervals of where $x$ can be.