Pointwise convergence.

Definition 10.1.1   Let $ (f(n)$ be a sequence of functions, all defined on the same domain $ \mathcal{D}$ . This sequence converges pointwise towards a function $ F$ , defined on $ \mathcal{D}$ too if, for any $ x \in \mathcal{D}$ , the sequence $ (f_n(x))$ converges towards $ F(x)$ , i.e.

$\displaystyle \forall x \in \mathcal{D}, \forall \varepsilon >0, \exists N_0 \; \vert \; n \geq N_0 \Longrightarrow \vert f(n(x)-F(x)\vert<\varepsilon.$    

Note that $ N_0$ depends not only on $ \varepsilon$ , but on the point $ x$ too.

Example 10.1.2  

Definition 10.1.3   Let $ (f(n)$ be a sequence of functions. The series of functionsConsider the sequence of functions $ f_n$ defined on the interval [0,1] by $ f_n(x)=x^n$ .

For every $ x \in [0,1)$ , the following holds:

$\displaystyle \underset{k=0}{\overset{n}{\sum}} f_k(x)=1+x+x^2+ \dots+ x^n = \frac {1-x^{n+1}}{1-x}$    

, Thus for $ 0 \leq x < 1$ , the sum of the series is given by

$\displaystyle S(x)=\frac{1}{1-x},$    

and at 1, the series is divergent (sum of a series with a non zero constant general term). $ \underset{k \geq 0}{\sum} f_n$ is pointwise convergent if the sequence of partial sums $ \underset{k = 0}{\overset{N}{\sum}} f_n$ is pointwise convergent.

Example 10.1.4  

Noah Dana-Picard 2007-12-28