Answer to Question #205259 in Calculus for Rajkumar

Question

using weierstrass m-test show that the following series converges uniformly∑ n=1 ∞ n^3 X^n,X€[-1/3,1/3]

Answer 1

Given,$\sum_{n=1}^{\infty} n^3x^n, x\in[-\frac{1}{3},\frac{1}{3}].\\ Then,\\ f_n=n^3x^n\to\infty as n\to \infty\\ i.e., \nexists M_n \\ \text{such that,}\\ f_n(x)|\leq M_n, \forall n\in \N, \forall x\in[-\frac{1}{3},\frac{1}{3}].\\ \text{Thus, by weierstrass m-test the given series is not uniformly covergent.}\\$