The Fundamental Theorem of Calculus
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Criteria for convergence
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(1) A necessary and sufficient condition for convergence is that there exists K such that [Equation goes here - download the original to see it.] for all N. This means, [Equation goes here - download the original to see it.] (provided [Equation.] for all ) That is, a bounded, increasing series is convergent. (2)Comparison test [Equation.] The following additional tests for convergence build on the basis established by the comparison criterion. (3) Cauchy’s test for convergence Let [Equation.] Then [Equation.] Proof Let [Equation.]. Let r be chosen such that [Equation.]. Since l is limit of [Equation.] there exists an N such that for all [Equation.] [Equation.] Hence Equation.] This is a geometric series with hence converges to a limit; hence, by the comparison test [Equation.]. If [Equation.], then for all we have [Equation.] which implies [Equation.]. (4) D’Alembert’s test Let [Equation.] and let [Equation.]. Then [Equation.] Proof Let [Equation.]. Let r be chosen such that [Equation.]. Since l is limit of [Equation.] there exists an N such that for all [Equation.]. Therefore, [Equation.] Then [Equation.] where K does not depend on n. Now [Equation.] is a geometric series with [Equation.] hence converges to a limit; hence, by the comparison test [Equation.]. If [Equation.], then for all [Equation.] we have [Equation.] which implies [Equation.] and the series diverges. (5) Absolute convergence When terms alternate between positive and negative values the above tests do not apply and no conclusion about the convergence of the series can be drawn. However, there is another test that may be applied. Theorem If [Equation.] converges then so to does [Equation.]. Proof Let [Equation.] Then [Equation.] and the series [Equation.] contains those terms of [Equation.] that are positive or zero. Also [Equation.] and the series [Equation.] contains those terms of [Equation.] that are negative or zero. Hence [Equation.] Given [Equation.] converges, then [Equation.] converge. Hence [Equation.] converges. Definition When [Equation.] converges the series [Equation.]is said to be absolutely convergent. (6) Alternating series Let [Equation.] Then the alternating series [Equation.] converges to a limit l such that [Equation.]. Proof Let [Equation.] Then [Equation.] This implies that the odd and even sequences [Equation.] both converge to the same limit; likewise [Equation.] converges to this limit. Since [Equation.] The sum of this series must lie between these two numbers. (7) Conditional convergence If [Equation.]converges but [Equation.] diverges, then [Equation.] is said to converge conditionally. The series is said to converge conditionally, because the limit depends on the order in which the terms is taken. (8) Theorem If [Equation.]is absolutely convergent, then every series comprising the same terms in any order has the same limit. Proof 1. Suppose all the terms of the series [Equation.] are positive; that is [Equation.] for all n. Let [Equation.] represent a series formed from a permutation of the same terms as [Equation.]. Let [Equation.] Suppose [Equation.] is a term in [Equation.]; then it is also contained in [Equation.] for some m. This applies to every term in [Equation.]. Therefore, for some r, the series [Equation.] is contained in [Equation.]. Therefore, [Equation.]. By reversing the argument we can also show [Equation.]. Thus [Equation.], so the two series converge to the same limit. 2. Now let be any absolutely convergent series. Let [Equation.] Since [Equation.] converges, [Equation.] and [Equation.] are also convergent. Let [Equation] represent a series formed from a permutation of the same terms as [Equation] and define [Equation] and [Equation] accordingly. The series [Equation], [Equation] [Equation] and [Equation] all comprise positive terms, so by part (1) above [Equation] and [Equation] have the same limit; likewise [Equation] and [Equation] have the same limit. Then [Equation] and [Equation] converge to the same limit.
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Contents of
The Fundamental Theorem of Calculus
1 Text: The fundamental theorem of calculus
2 Fundamental Theorem of Calculus
3 Infinite series
4 Integration as the reverse of differentiation
5 Convergence and divergence of series
6 Integration as the Sum of Approximations
7 The dissection
8 Properties of infinite series
9 Criteria for convergence
10 Definition, dissection
11 Definition, upper and lower approximative sums
12 Definition, upper and lower integrals
13 Definition, integrable function
14 Theorem, continuous functions are integrable
15 Properties of the integral
16 The fundamental theorem of calculus
17 Theorem, evaluation of the definite integral
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