Sequences and Series of Functions; Uniform Convergence
This domain tests the difference between pointwise and uniform convergence and the theorems that license interchanging limits with continuity, integration, and differentiation. Power series and the Weierstrass M-test are the main computational tools.
Definitions
f_n → f pointwise on E if for each x and each ε > 0 there is N depending on both x and ε. Convergence is uniform if for each ε > 0 there is one N working for all x in E and all n ≥ N. Equivalently, sup_{x in E} |f_n(x) − f(x)| → 0. The uniform Cauchy criterion replaces pointwise comparison by |f_n(x) − f_m(x)| < ε uniformly in x. Weierstrass M-test: if |f_n(x)| ≤ M_n on E and ΣM_n converges, then Σf_n converges uniformly and absolutely.
Interchange theorems
A uniform limit of continuous functions is continuous. If f_n → f uniformly on [a, b] and each f_n is integrable, then ∫f_n → ∫f. If f_n → f pointwise, each f_n is differentiable, and f_n' → g uniformly, then f is differentiable with f' = g. Dini's theorem: monotone pointwise convergence to a continuous limit on a compact set is uniform.
Power series
For Σa_n(x − c)^n the radius is R = 1/limsup |a_n|^{1/n}. The series converges absolutely for |x − c| < R, diverges for |x − c| > R, and converges uniformly on compact subintervals of (c − R, c + R). Termwise differentiation and integration are valid inside the radius.
Worked example
Take f_n(x) = x^n on [0, 1]. Pointwise, f_n(0) = 0, f_n(1) = 1, and for 0 < x < 1, x^n → 0, so the limit is 0 on [0, 1) and 1 at 1, which is discontinuous. A uniform limit of continuous functions would be continuous, so convergence is not uniform. Directly, sup_{[0,1]} |x^n − f(x)| = 1 for every n, because values near 1 stay close to 1. On [0, r] with r < 1 the supremum is r^n → 0, so convergence is uniform there.
Traps
Interchanging limit and integral without uniformity; assuming a pointwise limit of continuous functions is continuous; differentiating termwise without uniform convergence of derivatives; expecting uniform convergence at the endpoints of the interval of convergence; using the M-test with bounds that depend on x.