Additional discussion on the " Monster of Math"
(Weierstrass), after reading the message.
The ratio test:
|4-3|/2=1/2<1 (independent of x).
=> the series converges absolutely.
In fact, for n=1 to infinity,
SIGMA[1/2^n]. By the comparison test,
the latter is a G.P. with the ratio=1/2 is
the series is uniformly convergent with x
over D(-inf., inf.), as n->infinity.
Thus, we can write
But, f(x), strictly speaking is a function of x
but it is not an injection & contains the
"many to one" types of corresponding. For
any x1& x2 in D, if
for some k & n, but not exactly x1=x2.
The graph of it for n=20, is very complicated
like the "sound wave" but is continuous over
there. It may expect that f(x) is continuous
over D too.