Calculate right limit with Wolfram|Alpha?

Questions
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1

Hey,

limit of (1+((-1)^n - Wolfram|Alpha)%2Fn)%5En+as+n+approaches+infinity

Gives the limit as 1, when the sequence doesn’t actually converge. Or am I missing something?

All the best,
Sam

2

WolframAlfa give wrong answer. It should be: Exp[-1] to Exp[1] or :

Interval[{Exp[-1],Exp[1]}]

but Mathematica 11.0 gives No answer.

3

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]