Complex Sequences

Complex Analysis · Axiom Academy

When does an endless list of complex points settle on a single limit? Three pictures that make convergence in the plane obvious. 1. Convergence Is a Disk You Can't Escape Pick a target L and draw a disk of radius around it — as small as you like. The sequence exactly when, no matter how tiny that disk is, every term past some point N lands inside it and stays . The points may circle and zig-zag, but the tail is eventually trapped. Convergence, stated with distance The – N definition: stay inside every disk 2. A Complex Sequence Is Two Real Sequences Write and L = a + ib . Then if and only if the real parts and the imaginary parts . A single complex limit splits cleanly into two ordinary real limits you already know how to take. For , the real part is x_n = 1 for every n — already at its limit a = 1 . The imaginary part is . So b = 0 , and together the point heads for 1 + 0i . Each part is no bigger than the whole distance: and . Conversely , so if both parts vanish, so does the distance. Each point z_n drops a vertical thread to the real axis (landing on x_n = 1 ) and a horizontal thread to the imaginary axis (landing on y_n = 1/n ). Watch both shadows settle: the real shadow is pinned at 1 , the imaginary shadow slides down to 0 . Two real limits, one complex limit . 3. The Modulus Test: Watch the Distance Collapse

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