My Attempt: There is no dought that $$\sum_{1}^{\infty }\frac{1}{((n-1)x+1)(xn+1)}$$ converges pointwise on $[1,\infty)$ But What's about uniform convergence and uniform limit of sequence $$< \frac{1}{((n-1)x+1)(xn+1)}>$$ on $[1,\infty)$
2026-02-24 10:42:14.1771929734
$\sum_{1}^{\infty }\frac{1}{((n-1)x+1)(nx+1)}$ converges uniformly on $[1,\infty)$?
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Just compare the series with $\sum \frac 1 {((n-1)+1) (n+1)}$ which is convergent. By M-test the series is uniformly convergent.