Differentiation of Sequence of Functions

Definition

Differentiation of Sequence of Functions

Let $f:=\underset{n\to \infty }{\lim }{f}_n$

• If $(f_n)$ is a sequence of differentiable function on a bounded interval $I$,
• $\forall x_0\in I,\exists \underset{n\to \infty }{\lim }{f}_n(x_0)\wedge (|f(x_0)|<\infty )$, and
• $(f_n^{\prime })$ converges uniformly on the $I$,

Then $(f_n)$ converges uniformly on the interval $I$, the limit function $f$ is differentiable on $I$, and $\forall x_0\in I,f^{\prime }(x_0)=\underset{n\to \infty }{\lim }{f}_n^{\prime }(x_0)$

Differentiation of Series of Functions

Let $f:=\sum _{n=1}^{\infty }{f}_n$

• If $(f_n)$ is a sequence of differentiable function on a bounded interval $I$,
• $\forall x_0\in I,\exists \sum _{n=1}^{\infty }{f}_n(x_0)\wedge (|f(x_0)|<\infty )$, and
• $\sum _{n=1}^{\infty }{f}_n^{\prime }$ converges uniformly on the $I$,

Then $f$ converges uniformly, and differentiable on the interval $I$, and $\forall x_0\in I,f^{\prime }(x_0)=\sum _{n=1}^{\infty }{f}_n^{\prime }(x_0)$