ANSWER The series ∑n=1∞n4+x3nx is uniformly convergent on [0,α] .
EXPLANATION.
Since a) 0≤nx≤nα
b) n4+x3≥n4
for x∈[0,α] and n≥1 , then
0≤n4+x3nx≤n4nα=n3α .
The series ∑n=1∞n3α=α∑n=1∞n31 is convergent , because the series ∑n=1∞n31 is a p-series for p=3 .
Therefore, by the Weierstrass M-Test the series ∑n=1∞n4+x3nx is uniformly convergent on [0,α] .
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