Functions from Series

Real Analysis · Axiom Academy

A whole function, built out of nothing but powers of x : . You can build a function out of an infinite sum of powers In algebra a polynomial only ever has finitely many terms, so it can only bend so much. But let the sum run forever — — and something remarkable happens: the right coefficients make that endless sum of simple powers settle into the exact shape of a smooth curve. The infinite sum doesn't just approximate a function; it is one. Watch the chunky low-degree polynomials grow. Each new term bends the curve a little more; the degree-1, degree-2, degree-3, … partial sums of e^ x 's power series stack up and close in on the true e^ x curve until you can't tell them apart. More terms, less gap: the polynomial of every higher degree hugs e^ x tighter, and in the limit it lands exactly on the curve. Crank up the terms and watch a polynomial become Here is one of the cleanest examples in all of analysis: . Slide to add terms and watch the partial-sum polynomial climb out of the flat line and wrap itself onto the curve of . At x=0.6 the true value is ; every term you add pushes the running sum closer to it. The series only builds the function where it converges A power series doesn't define its function everywhere — only on an interval of convergence . Drag the point x along the axis. Inside the band |x|<1 the partial sums of pile up neatly on ; step outside it and the same terms blow up, refusing to settle on anything.

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