Question #303585

Using wiestrass M test show that the following series converges uniformly Sigma infinity n=1 n^3 x^n x belongs to[-1/3,1/3]

1
Expert's answer
2022-03-01T04:12:00-0500

The series

n=1n3xnx (1)\quad \quad \quad \quad \quad \quad \sum _{ n=1 }^{ \infty }{ { n }^{ 3 }{ x }^{ n }{x} } \quad \quad \quad \quad \quad \quad \quad \quad \quad \ (1)

is a power series , the radius of convergence of which is 

r= limnn3(n+1)3=1r=\ \lim _{ n\rightarrow \infty }{ \frac { { n }^{ 3 } }{ { (n+1) }^{ 3 } } } =1

 n1andx[13,13]: 0n3xnxn33n  (2)\forall \ n\ge 1\quad and\quad \forall x\in \left[\frac{-1}{3},\frac { 1 }{ 3 } \right] :\ 0\le { n }^{ 3 }{ x }^{ n }{x}\le \frac { { n }^{ 3 } }{ { 3 }^{ n } } \ \quad \quad \quad \quad \ \quad (2)

The series (1) converges in the interval (1,1)(-1,1). The point x=13x=\frac{1}{3} belongs to (1,1)(-1,1). Consequently series

n=1n23n (3)\quad\quad \quad\quad\quad\quad\quad\sum _{ n=1 }^{ \infty }{ \frac { { n }^{ 2 } }{ { 3 }^{ n } } } \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad\quad\quad\quad\ (3)

 

is convergent. According to the Weierstrass' M-Test , the series​ n=1n3xnx\sum _{ n=1 }^{ \infty }{ { n }^{ 3 }{ x }^{ n }{x} }

 converges uniformly in the interval [13,13]\left[ \frac{-1}{3},\frac { 1 }{ 3} \right] .

Option 2) The convergence of series (3) can be proved by the d'Alembert criterion 

an:=n23n,limnan+1an=limn(n+1)23n+13nn2=13<1{ a }_{ n }:=\frac { { n }^{ 2 } }{ { 3 }^{ n } } ,\lim _{ n\rightarrow \infty }{ \frac { { a }_{ n+1 } }{ { a }_{ n } } } =\lim _{ n\rightarrow \infty }{ \frac { { (n+1) }^{ 2 } }{ { 3 }^{ n+1 } } \cdot \frac { { 3 }^{ n } }{ { n }^{ 2 } } } =\frac { 1 }{ 3 } <1

Given the fulfillment of (2), according to the Weierstrass' M-Test, we obtain the series (1) converges uniformly in the interval  [13,13][\frac{-1}{3},\frac{1}{3}] .


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