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We have the inequality αn(t)4π(3π)1/3exp{t0(1+3FP(σ))dσ}t0P(σ)2(3CD21)1/3αn1(σ)dσ for n=2,3,. (We notice that αn appears on both sides of the inequality.)

Why does it follow that the infinite series n=1αn(t) converges locally uniformly on R+0 (that is, converges on [0,T] for 0tT)?

This comes from page 354 of the journal that contains the paper "Global symmetric solutions of the initial value problem of stellar dynamics" by Jurgen Batt.

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Some estimates like αn(s)A(T)n/n! for all s[0,T] should hold by induction, if we suppose something moderate on your functions, which depend on σ. This yields desired uniform convergence.

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  • As soon as we bounded αn(t) by Mn:=A(T)n/n! for all 0tT, we invoked the Weierstrass M-test to assert local uniform convergence of the infinite series, right? (I know, basic question, sorry)
    – cupcake
    Commented Apr 20, 2016 at 22:11
  • Yes, Weiersteass test, exactly. Commented Apr 20, 2016 at 22:16

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