Abel's Theorem and Weierstrass Approximation

Abel's theorem: continuity up to the endpoint

Inside its radius a power series is continuous because it converges uniformly on closed subintervals. But what happens exactly at an endpoint where it just barely converges? Abel's theorem says: if a power series with radius R = 1 converges at the endpoint x = 1, then its sum is continuous from the left there, so the value at the endpoint equals the limit of the sum as x -> 1 from below.

Why it is not free: at the endpoint the convergence may be only conditional, so the M-test gives no uniform convergence there and the easy continuity argument breaks. Abel's proof uses summation by parts to extract just enough uniformity on [0, 1] to push continuity to the boundary. The payoff is that we may legitimately plug the endpoint into series identities proven only inside the radius.

Recall from guide 3 (valid for |x| < 1):
   ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ...

Question: what is the value at the endpoint x = 1?

Step 1.  At x = 1 the series becomes  1 - 1/2 + 1/3 - 1/4 + ...,
         the alternating harmonic series.  Terms 1/n decrease to 0,
         so by the alternating series test it CONVERGES (to some S).

Step 2.  By Abel's theorem the power-series sum is continuous from
         the left at x = 1, so
            S = lim_{x -> 1^-} ln(1 + x) = ln 2.

Conclusion:  1 - 1/2 + 1/3 - 1/4 + ... = ln 2.

Abel licenses the limit x -> 1^- even though convergence at x = 1
is only conditional, not absolute.

Weierstrass: polynomials approximate everything continuous

The track has built functions from polynomials; the Weierstrass approximation theorem runs the arrow the other way. It states: if f is a continuous function on a closed bounded interval [a, b], then for every epsilon > 0 there is an ordinary polynomial p with sup over [a, b] of |f(x) - p(x)| < epsilon. In words, the polynomials are dense in the space of continuous functions under the sup norm.

The standard constructive proof uses Bernstein polynomials. To approximate f on [0, 1], form B_n(x) = sum from k=0 to n of f(k/n) * C(n,k) x^k (1-x)^(n-k). These are honest polynomials built only from sampled values of f, and they converge uniformly to f. The probabilistic heart is that the binomial weights C(n,k) x^k (1-x)^(n-k) concentrate near k/n ≈ x, so B_n(x) is a weighted average of f-values clustered around f(x).

The big picture and a generalization

1. Power/analytic functions form a narrow, rigid class: equal to their own Taylor series, hence determined by germ data at one point.
2. Continuous functions are vastly more general — most are nowhere analytic, some even nowhere differentiable.
3. Weierstrass bridges them: even though a continuous f need not BE a power series, it is a uniform LIMIT of polynomials on each compact interval.