Return to Answer

Also, you can change the interpolation-function partsumtrans to get a different performance.

Or like @cvgmt's:

DynamicModule[{t},
    Column[{Dynamic@Plot[
        Evaluate[{partsumtrans[f, T, 0, 5, 1][t][x]}~Join~
        (partsum[f, T, #][x]&/@Range[0,5])],
        {x, -1, 1},
        PlotRange->{{-1, 1}, {-1.2, 1.2}},
        AspectRatio->Full,
        PlotStyle->{
            Red, 
            Sequence@@(
                Directive[
                    Thick, Opacity[.5], 
                    Dashed, ColorData[1][#]
                ]&/@Range[0, 5]
            )
        },
        ImageSize->Medium,
        PlotLegends->LineLegend[
            {"animate", Sequence@@(ToString@#&/@Range[0, 5])}
        ]
    ], Animator[Dynamic[t], 
                AnimationRunning->False, 
                AppearanceElements->{
                "PlayButton", "PauseButton", "ResetButton"}]},
        Alignment->Center           
    ]
]

Also, you can change the interpolation-function partsumtrans to get a different performance.

And you can use them to construct a demonstration:

Also, you can change the interpolation-function partsumtrans to get different performance.

Now we can use them to construct a demonstration:

Also, you can change the interpolation-function partsumtrans to get a different performance.

@Bob Hanlon Style:

Show[ParametricPlot3D[
    Evaluate[{x, 2#/10, partsum[f, T, #][x]}&/@Range[0, 10]],
    {x, -1, 1},
    PlotStyle->Thick
],
ParametricPlot3D[
    {x, y, partsumtrans[f, T, 0, 10, 2][y][x]},
    {x, -1, 1}, {y, 0, 2},
    Exclusions->None,
    Mesh->None, PlotStyle->Directive[Opacity[.5]],
    ColorFunction->"Rainbow",
    PlotPoints->30
]]

Also, you can change the interpolation-function partsumtrans to get different performance.