The Fundamental Theorem of Calculus
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Properties of infinite series
Equations are omitted for technical reasons - download the original pdf
(1) The convergence or divergence of a series is unaffected if a finite number of terms are inserted, or suppressed, or altered. (2)If [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (3) If [Equation goes here - download the original to see it.] This makes [Equation goes here - download the original to see it.] a necessary condition of convergence. However, it is not sufficient. (4) [Equation goes here - download the original to see it.] is convergent , then so is any series whose terms are obtained by bracketing the terms of [Equation goes here - download the original to see it.] in any manner, and the two have the same sum. This means that we can add brackets into a convergent series but the converse is not generally true; we cannot indiscriminately remove them. For example, [Equation goes here - download the original to see it.] oscillates, but the series [Equation goes here - download the original to see it.] converges to zero. (5) If [Equation goes here - download the original to see it.] for every n, then [Equation goes here - download the original to see it.] either converges or diverges to [Equation].
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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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