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6 Exercises 1. Find limn→∞ n 3n+4 . 2. Find limn→∞ 3n4 +7n+1000 . n4 +1 3. Find limn→∞ 2n +7(5n ) 4n +2(5n ) . 4. Find limn→∞ (n2 + 6n) − n. Hint: Multiply and divide by 5. Find limn→∞ n 1 k=1 10k . (n2 + 6n) + n. 6. Suppose {xn + iyn } is a sequence of complex numbers which converges to the complex number x + iy. Show this happens if and only if xn → x and yn → y. n k k=0 r . 7. 9. Hint: First show n k k=0 r 8. Using the binomial theorem prove that for all n ∈ N, 1 + Hint: Show first that n k = n·(n−1)···(n−k+1) .

Similarly there is a general notion of what it means to be open. 3 Let U be a set of points. A point, p ∈ U is said to be an interior point if whenever |x − p| is sufficiently small, it follows x ∈ U also. The set of points, x which are closer to p than δ is denoted by B (p, δ) ≡ {x ∈ F : |x − p| < δ} . This symbol, B (p, δ) is called an open ball of radius δ. Thus a point, p is an interior point of U if there exists δ > 0 such that p ∈ B (p, δ) ⊆ U . An open set is one for which every point of the set is an interior point.

In mathematics, you can sometimes show certain things do not exist. It is very seldom you can do this outside of mathematics. For example, does the Loch Ness monster exist? Can you prove it does not? 4I will present a treatment of the trig functions which is independent of plane geometry a little later. 1 Basic Definitions A set is a collection of things called elements of the set. For example, the set of integers, the collection of signed whole numbers such as 1,2,-4, etc. This set whose existence will be assumed is denoted by Z.

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Conditional logistic analysis of case-control studies with complex sampling by Langholz B., Goldstein L.


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