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Calculus Deconstructed: A Second Course in First-Year by Zbigniew H. Nitecki

By Zbigniew H. Nitecki

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22. x0 = 1, xn := xn−1 xn−1 +1 , n = 1, ... 23. x0 = 2, xn := xn−1 − 1 xn−1 , n = 1, ... 24. x0 = 1, xn := xn−1 + 1 xn−1 , n = 1, ... 25. x0 = x1 = 1, xn = xn−2 + xn−1 for n = 2, ... ) 26. x0 = 0, x1 = 1, xn = 12 (xn−2 + xn−1 ), n = 2, ... 27. x0 = 0, x1 = 1, xn = 12 (xn−2 − xn−1 ), n = 2, ... 2 Limits of Real Sequences How do we “know” a number? Counting is good, and using this we can get any whole number. Ratios are almost as good: by dividing the space between two integers into equal parts and then counting, we can locate any fraction precisely on the number line.

Hint: Try a few terms, maybe rewrite the formula, or think geometrically! ) (d) Show that this sequence converges. (e) You can probably guess what the limit appears to be. Can you prove that your guess is correct? 1 34. Show that the harmonic series ∞ k=1 k diverges to infinity, as follows (a version of this proof was given by Nicole Oresme (1323-1382) in 1350 [20, p. 91] and rediscovered by Jacob Bernoulli (1654-1705) in 1 1689 [51, pp. 320-4]): let SK := K k=0 k be the partial sums of the harmonic series.

A) Show that the square of an even integer is even. (b) Show that the square of an odd integer is odd. (Hint: A number q is odd if it can be written in the form q = 2n + 1 for some integer n. Show that then q 2 can also be written in this form. ) (c) Show that if m is an integer with m2 odd (resp. even), then m is odd (resp. even). 31. Find an example of a convergent sequence {xk } contained in the open interval (0, 1) whose limit is not contained in (0, 1). (Hint: Look at small fractions. ) 32.

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