Answer to Question #212324 in Calculus for Vikas

Question

9.a) Check whether the series sum_(m=1)^(oo)(n^(2)x^(5))/(n^(4)+x^(3)) x in 0 alpha is uniformly convergent or not where alpha in R^(+).

Answer 1

ANSWER. The series $\sum _{ n=1 }^{ \infty }{ \frac { { n }^{ 2 }{ x }^{ 5 }\ }{ { n }^{ 4 }+{ x }^{ 3 } } }$ is uniformly convergent in $\left[ 0,\alpha \right]$ .

EXPLANATION. We use Weierstrass M-Test to prove this

Since $x\in \left[ 0,\alpha \right]$, then

$0\le \frac { { n }^{ 2 }{ x }^{ 5 }\quad }{ { n }^{ 4 }+{ x }^{ 3 } } \le \frac { { n }^{ 2 }{ \ \alpha }^{ 5 }\ }{ { n }^{ 4 }\quad }$ .

Denote ${ f }_{ n }(x)=\frac { { n }^{ 2 }{ x }^{ 5 }\ }{ { n }^{ 4 }+{ x }^{ 3 } } ,\ { M }_{ n }=\frac { { \ \alpha }^{ 5 } }{ { n }^{ 2 } }$ . We have $\left| { f }_{ n }(x) \right| ={ f }_{ n }(x)\le { M }_{ n }$ and the series $\sum _{ n=1 }^{ \infty }{ { M }_{ n }\ } ={ \alpha }^{ 5 }\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 2 } } }$ converges (it is p-series , p=2). So, by the Weierstrass M-Test the series $\sum _{ n=1 }^{ \infty }{ { f }_{ n }(x)= } \sum _{ n=1 }^{ \infty }{ \frac { { n }^{ 2 }{ x }^{ 5 }\ }{ { n }^{ 4 }+{ x }^{ 3 } } }$ converges uniformly in $\left[ 0,\alpha \right]$ (α>0).