Uniform Convergence

Question

How do we show that sigma(n=1 to infinity){ [1/[ n^p+ (n^q*x²) )] } is uniformly convergent , where x not equal to 0 & when p>1

Types of tests that can be used: Weierstrass M-test, Cauchy Criterion for uniform convergence, M_n test etc.

Answer 1

This a direct consequence of the Weierstrass M-test. Indeed, 1/[n^p + n^q x²] ≤ 1/n^p for all x, and ∑[n = 1, ∞] 1/n^p < ∞ since p > 1. By the Weierstrass M-test, the series ∑[n=1, ∞] 1/[n^p + n^q x²] converges uniformly on (-∞,∞).