(a) Give a proof of Weierstrass M-test.

(b) Derive Theorem 4 from Theorem 3.

(c) Prove that uniform convergence of a series in a region G implies uniform convergence in any portion of G. Is the converse true?

(d) Find the precise region of convergence of the series in Example 2 with x replaced by a complex variable z.

(e) Show that x² Σ^ˆž_m=1 (1 + x²)^-m = 1 if x ‰  0 and 0 if x = 0. Verify by computation that the partial sums s₁, s₂, s₃ look as shown in Fig. 369

Transcribed Image Text:

82 81 -1 х