using weierstrass m-test show that the following series converges uniformly∑ n=1 ∞ n^3 X^n,X€[-1/3,1/3]
Given,∑n=1∞n3xn,x∈[−13,13].Then,fn=n3xn→∞asn→∞i.e.,∄Mnsuch that,fn(x)∣≤Mn,∀n∈N,∀x∈[−13,13].Thus, by weierstrass m-test the given series is not uniformly covergent.\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.}\\∑n=1∞n3xn,x∈[−31,31].Then,fn=n3xn→∞asn→∞i.e.,∄Mnsuch that,fn(x)∣≤Mn,∀n∈N,∀x∈[−31,31].Thus, by weierstrass m-test the given series is not uniformly covergent.
