(Subscript[a, k_]):=(Piecewise[{{(((-1))^((k-1)/2))/k!, OddQ[k]}, {0, EvenQ[k]}}]);
Plot[Evaluate@Append[
Table[Labeled[(∑_(k=0)^n)(Subscript[a, k])(x^k),n], {n,{1,3,5,7}}],
Labeled[Sin[x],Sin]
], {x,-2π,2π}, PlotRange ->{-1.5,1.5}]

Column@Table[k->N[(Subscript[a, k])(157^k)],{k,{0,3,9,13,29,35,157,223,457}}]

{N[(∑_(k=0)^457)(Subscript[a, k])(157^k)],(∑_(k=0)^457)N[(Subscript[a, k])(157^k)]}

N@Sin[157]

w=N@WolframAlpha["Width of a Hydrogen Atom", "Result"]

ϕ[q_] := (For[k=0,Abs[(Subscript[a, k])(157^k)]<q/w,++k]; {q,q/w,k})

ϕ@Quantity[0.00078 0.01, "Meters"]

ϕ@Quantity[553.3, "Meters"]

Diameter of the rings of Saturn:

ϕ@Quantity[280000 1000, "Meters"]

ϕ[Quantity[100000×365×24×3600×300000×1000, "Meters"]]

ϕ[Quantity[93×(10^9)×365×24×3600×300000×1000, "Meters"]]