Riemann Integrability of the Sum of a Uniformly Con. Ser. of Functs.

# Riemann Integrability of the Sum of a Uniformly Convergent Series of Functions

On the Riemann Integrability of the Limit Function of a Uniformly Convergent Sequences of Functions page we saw that if $(f_n(x))_{n=1}^{\infty}$ is a sequence of real-valued functions with common domain $[a, b]$ that is uniformly convergent to the limit function $f(x)$ then $f$ is Riemann-integrable on $[a, b]$ and:

(1)
\begin{align} \quad \int_a^b f(x) \: dx = \int_a^b \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_a^b f_n(x) \: dx \end{align}

We will now look at analogous result for the Riemann integrability of the sum of a uniformly convergent series of Riemann integrable functions.

 Theorem 1: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of real-valued functions with common domain $[a, b]$. If the series of functions $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to the sum function $f$ then $\displaystyle{\int_a^b f(x) \: dx = \int_a^b \sum_{n=1}^{\infty} f_n(x) \: dx = \sum_{n=1}^{\infty} \int_a^b f_n(x) \: dx}$.
• Proof: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of real-valued functions with common domain $[a, b]$ and suppose that $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to the sum function $f(x)$. Let $(s_n(x))_{n=1}^{\infty}$ be the sequence of partial sums for this series. Then we have that $(s_n(x))_{n=1}^{\infty}$ uniformly converges to $f(x)$.
• Now, for each $n \in \mathbb{N}$ we have that:
(2)
\begin{align} \quad s_n(x) = \sum_{k=1}^n f_n(x) = f_1(x) + f_2(x) + ... + f_n(x) \end{align}
• So $s_n$ is a finite sum of Riemann integrable functions on $[a, b]$, so $s_n$ is Riemann integrable on $[a, b]$. But then we have that:
(3)
\begin{align} \quad \int_a^b f(x) \: dx = \int_a^b \sum_{n=1}^{\infty} f_n(x) \: dx &= \lim_{n \to \infty} \int_a^b s_n(x) \: dx \\ \quad &= \lim_{n \to \infty} \int_a^b \sum_{k=1}^{n} f_k(x) \: dx \\ \quad &= \lim_{n \to \infty} \int_a^b [ f_1(x) + f_2(x) + ... + f_n(x) ] \: dx \\ \quad &= \lim_{n \to \infty} \int_a^b f_1(x) \: dx + \int_a^b f_2(x) \: dx + ... + \int_a^b f_n(x) \: dx \\ \quad &= \sum_{n=1}^{\infty} \int_a^b f_n(x) \: dx \quad \blacksquare \end{align}