The series
∑n=1∞n3xnx (1)
is a power series , the radius of convergence of which is
r= limn→∞(n+1)3n3=1
∀ n≥1and∀x∈[3−1,31]: 0≤n3xnx≤3nn3 (2)
The series (1) converges in the interval (−1,1). The point x=31 belongs to (−1,1). Consequently series
∑n=1∞3nn2 (3)
is convergent. According to the Weierstrass' M-Test , the series ∑n=1∞n3xnx
converges uniformly in the interval [3−1,31] .
Option 2) The convergence of series (3) can be proved by the d'Alembert criterion
an:=3nn2,limn→∞anan+1=limn→∞3n+1(n+1)2⋅n23n=31<1
Given the fulfillment of (2), according to the Weierstrass' M-Test, we obtain the series (1) converges uniformly in the interval [3−1,31] .
Comments