Question

10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r < R then the series converges uniformly on (z: z - a s r) Proof. We may suppose that a = 0. If |z| < R there is an r with Izl < r < R Thus, there is an integer N such that |a,|1/" for al n 2 Nbecause - > a-:"I < ) for all ,12 N. This says that F, and since 1 Now suppose r < R and choose ρ such that r < ρ < R. As above, let N be such that anl <-n for all n N. Then if Iz! r, lanz"| |-) and series diverges; Moreover, the number R is the only number having properties (a) and (b) But then la,! <-and so la/ < (-1 ) for all n the tail end Σ anz" is dominated by the series the power series converges absolutely for each z< R. < 1 . Hence the Weierstrass M-test gives that the power series converges r). This proves parts (a) and (c). uniformly on (z: z| s

Answer 1

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