Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Propositions from Power series

All propositions from Power series.

Back to Power series

Pointwise convergence is not enough

Convergence at each fixed xx (called pointwise convergence) does not, by itself, justify differentiating the limit term by term.

Consider

fn(x)=sin⁡(n2x)n.f_n(x)=\frac{\sin(n^2x)}n.

For every fixed xx,

∣fn(x)∣≤1n⟶0,|f_n(x)|\le\frac1n\longrightarrow0,

so fn(x)→f(x)=0f_n(x)\to f(x)=0. However,

fn′(x)=ncos⁡(n2x),fn′(0)=n,f_n'(x)=n\cos(n^2x), \qquad f_n'(0)=n,

which does not converge to f′(0)=0f'(0)=0. In general,

lim⁡n→∞lim⁡h→0fn(x+h)−fn(x)handlim⁡h→0lim⁡n→∞fn(x+h)−fn(x)h\lim_{n\to\infty}\lim_{h\to0} \frac{f_n(x+h)-f_n(x)}h \quad\text{and}\quad \lim_{h\to0}\lim_{n\to\infty} \frac{f_n(x+h)-f_n(x)}h

need not agree. Power-series theorems supply the extra control inside the radius of convergence.

  1. [1]:Differentiate and integrate term by term

On this page

Pointwise convergence is not enough