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Here is what it looks like

sage: def f(x): 
....:     return sum(sin(k*x)/(k**2+1) for k in range(1000))                                                                     
sage: plot(f,0,2*pi)

Not clear if this is a known function.

Here is what it looks like

sage: def f(x): 
....:     return sum(sin(k*x)/(k**2+1) for k in range(1000))                                                                     
sage: plot(f,0,2*pi)

Not clear if this is a known function.