Question

Use Weierstrass’ M-Test to prove that the series ∑(x^nXn^2)
∞
n=1

n converges uniformly in x
the interval .
(0,1/5)

Answer 1

EXPLANATION

For all $x\in \left( 0,\frac { 1 }{ 5 } \right)$ and $n\ge 1\quad$

$0<{ n }^{ 2 }{ x }^{ n }<\frac { { n }^{ 2 } }{ { 5 }^{ n } }$ (1)

$\lim _{ n\ \dashrightarrow \infty }{ \sqrt [ n ]{ \frac { { n }^{ 2 } }{ { 5 }^{ n } } } } =\lim _{ n\dashrightarrow \infty }{ \frac { { \left( \sqrt [ n ]{ n } \right) }^{ 2 } }{ 5 } =\frac { 1 }{ 5 } <1\quad }$$\left( \lim _{ n\rightarrow \infty }{ \sqrt [ n ]{ n } =1 } \right)$

Consequently , by Cauchy root test the series

$\sum _{ n=1 }^{ \infty }{ \frac { { n }^{ 2 } }{ { 5 }^{ n } } \quad \quad \quad \ }$ (2)

converges.

(1) and the convergence of series (2) means that the conditions of Weierstrass’ M-Test are satisfied. Therefore, the series $\sum _{ n=1 }^{ \infty }{ { n }^{ 2 } } { x }^{ n }\quad$converges uniformly on the set $\left( 0,\frac { 1 }{ 5 } \right)$