Plot[(∑_(i=1)^100)(Cos[i x]),{x,-0.1,0.1},PlotRange->All]

Plot[(∑_(i=1)^700)(Cos[i x]),{x,-0.1,0.1},PlotRange->All]

Convert function to points Plot->ListPlot:

ftq=Table[(∑_(i=1)^700)(Cos[i x]),{x,-0.1,0.1, 0.001}];

ListPlot[ftq,Joined->True,PlotRange->All]

Now FT (NOTE because of the math symmetry, take 1/2 out

ListPlot[InverseFourier[Fourier[ftq]],PlotRange->All,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->True,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->True,Joined->True]

ListPlot[Abs[Fourier[ftq]],PlotRange->{{0,100},{0,300}},Joined->True]

Plot[(∑_(i=1)^700)(Sin[i x]),{x,0,0.1},PlotRange->All]

Plot[(∑_(i=1)^700)(i Sin[i x]),{x,0,0.1},PlotRange->All]

((∑_(i=1)^700)(i Sin[i x]))*Sin[i x]

(∫_(-Pi)^Pi)((∑_(n=1)^700)(n Sin[n x]))*Sin[10 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[x])*Sin[x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[x])*Sin[x]\[DifferentialD]x

∫ Sin[n x]*Sin[x]\[DifferentialD]x

∫ Sin[n x]*Sin[ m x]\[DifferentialD]x

Clear[m,n]

∫ Sin[3 x]*Sin[ 2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[  3 x])*Sin[2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Sin[  3 x])*Sin[3 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Cos[  3 x])*Cos[2 x]\[DifferentialD]x

(∫_(-Pi)^Pi)( Cos[  3 x])*Cos[3 x]\[DifferentialD]x