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- Working directly from definition, prove that if zn and wn are sequences of complex numbers withlimn→∞zn = 4 + 3i, and limn→∞wn = 4 − 3i,then limn→∞zn · wn = 25. (You may use the fact that convergent complex sequences are bounded.)44^2 + y^2 = z^2 Identify all (x, y,z) primitive Pythagorean triples such thatWhat is Central Limit Theorem? Please give a simple example with a line graph. Where will bethe use of Central Limit Theorem allows us to solve a complex problem. (at least 5 sentencesand graphs).
- Let Q^2 be the rational plane of all ordered pairs (x,y) of rational numbers with the usual interpretations of the undefined geometric terms used in analytical geometry. Show that axiom C-1 and the elementary continuity principle fail in Q^2. (Hint: the segment from (0,0) to (1,1) can not be laid off on the x axis from the origin. Axiom C-1: If A, B are two points on a line a, and if A′ is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅ AB.A. i. Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5) . (2n devided 3n+5..ii. Working directly from the definition of limits, give a direct verification that your answer in (i) is correct. (Your answer should involve the letter ε.)1. Draw two perspectivities so that a projectivity is established between A, B, C and A′, B′, C′. 2. Draw the axis of homology for the projectivity in the previous item with A corresponding to A′. B to B′ and C to C′. 3. Draw the image of X in the this projectivity.
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