By Frank Ayres

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**Extra resources for Schaum's Outline of Calculus, 6th Edition: 1,105 Solved Problems + 30 Videos (Schaum's Outlines)**

Instance four: (a) ; (b) ; (c) . word that, in case (c), the series converges neither to +∞ nor to –∞. instance five: The series (–1)n is divergent, however it diverges neither to +∞ nor to –∞ nor to ∞ Its values oscillate among 1 and –1. a series sn is expounded to be bounded above if there's a quantity c such that sn c for all n, and sn is expounded to be bounded lower than if there's a quantity b such that b sn for all n. a chain sn is expounded to be bounded whether it is bounded either above and less than. it really is transparent series sn is bounded if and provided that there's a quantity d such that |sn| d for all n. instance 6: (a) The series 2n is bounded under (for instance, by way of zero) yet isn't bounded above. (b) The series (–1)n is bounded. word that (–1)n is –1; 1; –1, . . . , So, |(–1)n|1 for all n. Theorem forty two. 1: each convergent series is bounded. For an explanation, see challenge five. The speak of Theorem forty two. 1 is fake. for instance the series (–1)n is bounded yet no longer convergent. average mathematics operations on convergent sequences yield convergent sequences, because the following intuitively seen effects express. Theorem forty two. 2: suppose . Then: (a) , the place ok is a continuing. (b) , the place ok is a continuing. (c) . (d) . (e) . (f) only if d ≠ zero and tn ≠ zero for all n. For proofs of components (c) and (e), see challenge 10. the next evidence approximately sequences are intuitively transparent. Theorem forty two. three: . For an explanation, see challenge 7. Theorem forty two. four: (a) . (b) . For proofs, see challenge eight. Theorem forty two. five (Squeeze Theorem): If , and there's an integer m such that sn tn un for all n m, then . For an evidence, see challenge eleven. Corollary forty two. 6: If and there's an integer m such that |tn| |un| for all n m, then . it is a outcome of Theorem forty two. five and the truth that is similar to . instance 7: . to work out this, use Corollary forty two. 6, noting that and . Theorem forty two. 7: imagine that f is a functionality that's non-stop at c, and think that , the place the entire phrases sn are within the area of f. Then See challenge 33. it truly is transparent that even if a series converges wouldn't be tormented by deleting, including, or changing a finite variety of phrases before everything of the series. Convergence relies on what occurs "in the lengthy run". we will expand the idea of endless series to the case the place the area of a series is authorized to be the set of nonnegative or any set including all integers more than or equivalent to a set integer. for instance, if we take the area to be the set of nonnegative integers, then 2n + 1 could denote the series of optimistic bizarre integers, and 1/2n may denote the series MONOTONIC SEQUENCES (a) a series sn is expounded to be nondecreasing if sn sn+1 for all n. (b) a chain sn is related to be expanding if sn sn+1 for all n. (c) a chain sn is related to be nonincreasing if sn sn+1 for all n. (d) a series sn is related to be reducing if sn > sn+1 for all n. (e) a series is related to be monotonic whether it is both nondecreasing or nonincreasing. basically, each expanding series is nondecreasing (but no longer conversely), and each lowering series is nonincreasing (but no longer conversely).