Download e-book for iPad: Calculus Deconstructed: A Second Course in First-Year by Zbigniew Nitecki

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41 and so on, form a sequence {xk } in R (where xk has precisely k digits after the decimal point). We have chosen each element xk of√the sequence to be the highest k-digit decimal fraction which is less than 2, and so we can say that for k = 0, 1, 2, ... ). In particular, xk is closer to 2 than it is to the next decimal fraction, so √ xk − 2 < 10−k . √ This means, if we tried to use xk in place of 2 in some calculation, our input would be off by an amount (called the error) which we don’t know 5 We conveniently gloss over the possibility of an infinite string of successive 9’s, which can confuse matters, but only for pedants.

Notice that different formulas can give the same sequence: what matters is that the same numbers appear in the same order. A second notation places the formula for an element in terms of the index inside braces, usually with a subscript and superscript indicating the range of the index. The sequence of odd integers could be written {2j + 1j }∞ j=0 or {2j − 1j }∞ j=1 . ) These sequences were specified in closed form, telling us how to determine the value of each element directly from its index.

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. , the sum σn := 2n below by 2n+1 = 21 . 2n+1 −1 1 k=2n k (b) Use Exercise 31b to conclude that the series is bounded ∞ n=0 σn diverges. , so that the subsequence S2N of {SK } diverges to infinity. 4 1 = ∞. k Finding limits It is natural to expect that if we perform arithmetic operations on convergent sequences, these should carry over to the limit.

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Calculus Deconstructed: A Second Course in First-Year Calculus (Draft version) by Zbigniew Nitecki


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