Exercises on series of functions

1. Consider f_n(x) = nx/(1 + n²x²) and f(x) = 0. Show that

   1. f_n → f pointwise on ℝ, however the convergence is not uniform.

   2. f_n → f uniformly on {x : |x| > k > 0}.

   Solutions

   1. 1. f(1/n) = 1/2, hence the convergence is not uniform.

      2. We must show that

         |f_n(x) − f(x)| < ε  forl all n > n₀ and all x ∈ {x : |x| > k > 0}.

         or equivalently sup { |f_n(x) − f(x)| : x ∈ I } < ε ∀ n > n₀.

         |f_n(x) − f(x)| is |f_n(x) − 0|, and so we must study how the function |nx / (1 + n² x²)| behaves in the domain. The function is basically n|x| / (1 + n²x²). We have that

         n|x| / (1 + n²x²) < n|x| / (x²n²) = |x|/nx² = 1/n ⋅ 1/|x|

         given that |x| > k, 1/|x| < 1/k, so your |f_n(x)| < 1/n ⋅ 1/k, this is enough to prove that lim _{n ⟶ ∞} sup |f_n(x)| = 0. ■

      «Sequences of functions Index Sulfides»