I don't think the limit exists. If you plot the values the sequence alternates between an "upper" limit approaching e, and a lower limit approaching 1/e.

See the attached code and plots f[n_] := (1 + ((-1)^n)/n)^n; e[n_] := (1 + 1/n)^n DiscretePlot[{e[n], f[n]}, {n, 1, 50}, PlotStyle -> {{Red, PointSize[.045], Opacity[.25]}, {Blue, PointSize[.015], Opacity[.85]}}, PlotTheme -> "Detailed", PlotRange -> All] DiscretePlot[{e[n], f[n]}, {n, 10000, 10020, 2}, PlotStyle -> {{Red, PointSize[.045], Opacity[.25]}, {Blue, PointSize[.015], Opacity[.45]}}, PlotTheme -> "Detailed", PlotRange -> All] DiscretePlot[{e[n], f[n]}, {n, 10001, 10021, 2}, PlotStyle -> {{Red, PointSize[.045], Opacity[.25]}, {Blue, PointSize[.015], Opacity[.45]}}, PlotTheme -> "Detailed", PlotRange -> All]