Series of Functions

• A series of functions is an expression of the form $\sum _{n=1}^{\infty }{f}_n(x)$, where $(f_n)$ is a sequence of functions.
• The series $\sum _{n=1}^{\infty }{f}_n(x)$ converges pointwise (or uniformly) to $f$ on $I$ if the sequence of partial sums $(S_n)$ converges pointwise (or uniformly) to $f$ on $I$. (where $S_n(x)=\sum _{k=1}^n{f}_k(x)$).
• Examples of series of functions include power series, and Fourier series.

Theorems

Cauchy Criterion for UC

• (6.6*) $\sum _{n=1}^{\infty }{f}_n(x)$ converges uniformly on $I$, if and only if, $\forall \epsilon >0,\exists N\in \mathbb{N}:\forall n>N,\forall m>n,\forall x\in I,\left|\sum _{k=n+1}^m{f}_k(x)\right|<\epsilon$

Dini’s Theorem for Series

• (6.5*, Dini’s Theorem)
  • Given $\sum _{n=1}^{\infty }{f}_n(x)$ converges pointwise to $f$ on $[a,b]$, if:
    • Each $f_n$ is continuous and non-negative on $[a,b]$,
    • $f$ is continuous in $[a,b]$
  • Then $\sum _{n=1}^{\infty }{f}_n(x)$ converges uniformly to $f$ on $[a,b]$

Weierstrass M-Test

• (6.7, Weierstrass M-Test) if:
  • Let $(f_n)$ be a sequence of functions defined on $I$.
  • There exists a sequence $(M_n)$ s.t. $\forall n\in \mathbb{N},\forall x\in I,|f_n(x)|\le M_n$ (thus $0\le M_n$)
  • The series $\sum _{n=1}^{\infty }{M}_n$ converges.
  • Then:
    • The series of functions $\sum _{n=1}^{\infty }{f}_n(x)$ uniformly converges on $I$.
    • For all $x\in I$, the series $\sum _{n=1}^{\infty }{f}_n(x)$ converges absolutelytodo

(6.4*)

• Given a series of functions $\sum _{n=1}^{\infty }{f}_n(x)$ that uniformly converges on an interval $I$ to a function $f$, If each $f_n$ is continuous on $I$, then $f$ is continuous on $I$.

Term-by-Term Integration

• Given a series of functions $\sum _{n=1}^{\infty }{f}_n(x)$ that uniformly converges on an interval $I$ to a function $f$
  • (6.18*, term-by-term integration) If each $f_n$ is integrable on $I$, then $f$ is integrable on $I$ and ${\int }_a^{b}f(x)dx=\sum _{n=1}^{\infty }{\int }_a^b{f}_n(x)dx$.

Term-by-Term Differentiation

• (6.9*) (term-by-term differentiation)

  • If:
    • Each $f_n$ is continuously differentiable on $I$
    • There exists $x_0$ in which $\sum _{n=1}^{\infty }{f}_n(x_0)$ converges
    • $\sum _{n=1}^{\infty }{f}_n^{\prime }(x)$ converges uniformly on $I$
    • then:
      • $\sum _{n=1}^{\infty }{f}_n(x)$ converges uniformly on $I$ to a function $f$ that is differentiable on $I$
      • $\forall x\in I,f^{\prime }(x)=\sum _{n=1}^{\infty }{f}_n^{\prime }(x)$

• (6.9) Let $(f_n)$ be a sequence of continuously differentiable functions on $[a,b]$ that pointwise converges to a function $f$ on $[a,b]$. If $(f_n^{\prime })$ converges uniformly on $[a,b]$, Then, $f$ is differentiable on $[a,b]$, and $\forall x\in [a,b],f^{\prime }(x)=\underset{n\to \infty }{\lim }{f}_n^{\prime }(x)$.