Series of functions

Let {f_n} be a sequence of functions defined on I ⊆ ℝ. We introduce a new sequence {s_n}, where

$$s_n(x)=\sum _{k=1}^n{f}_k(x),x\in I$$

Here, s_n(x) is called the nth partial sum of the series of functions

$$\sum _{n=1}^{\mathrm{\infty }}{f}_n(x)$$

This series is said to converge pointwise (uniformly) on I to f(x) if {s_n} converges pointwise (uniformly) to f(x) on I as n → ∞. Further, the series of functions is said to converge absolutely to f̄(x) on I if ∑^∞_n=1 |f_n(x)| converges for each x ∈ I. The series of functions is uniformly bounded on I iff there exists a real constant L such that ∑^k=1_n ≤ L for all x ∈ I and n ∈ ℕ.

Example 8.3.1 Consider the series ∑^∞_n=1 f_n(x) where,

Clearly, ∑^∞_n=1 f_n(x) is a series of continuous functions f_n(x) on ℝ and since its nth partial sum is s_n(x) = nx /(1 + n²x²) from exercise 1 it follows that this series converges pointwise to f(x) = 0 on ℝ, but the convergence is not uniform.

«Sequences of functions Index Exercises on series of functions»