Question #319201

Check whether the series sum_(n=1)^(oo)(nx)/(n^(4)+x^(3)) x in 0 alpha is uniformly convergent or not

1
Expert's answer
2022-03-28T18:15:40-0400

ANSWER The series n=1nxn4+x3\sum_{n=1}^{\infty}\frac{nx}{n^{4}+x^{3}} is uniformly convergent on [0,α]\left [ 0, \alpha \right ] .

EXPLANATION.

Since a) 0nxnα0\leq nx\leq n \alpha

b) n4+x3n4n^{4}+x^{3}\geq n^{4}

for x[0,α]x\in[0,\alpha] and n1n\geq1 , then

0nxn4+x3nαn4=αn30\leq\frac{nx}{n^{4}+^x{3}}\leq\frac{n\alpha}{n^{4}}=\frac{\alpha}{n^{3}} .

The series n=1αn3=αn=11n3\sum_{n=1}^{\infty}\frac{\alpha}{n^{3}}=\alpha\sum_{n=1}^{\infty }\frac{1}{n^{3}} is convergent , because the series n=11n3\sum_{n=1}^{\infty }\frac{1}{n^{3}} is a pp-series for p=3p=3 .

Therefore, by the Weierstrass M-Test the series n=1nxn4+x3\sum_{n=1}^{\infty}\frac{nx}{n^{4}+x^{3}} is uniformly convergent on [0,α]\left [ 0, \alpha \right ] .


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