The Fundamental Theorem of Calculus
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Convergence and divergence of series
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A bounded, increasing series is convergent. Addition and multiplication of sums of infinite series. Sums of series can be added and multiplied together. This is because they are limits of sequences so inherit those properties from sequences. Divergence A series that is not convergent is called divergent. Divergence series might oscillate or diverge to [Equation goes here - download the original to see it.] Examples (1) Geometric series [Equation goes here - download the original to see it.] Proof [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The series converges only if [Equation goes here - download the original to see it.] (2) The harmonic series Let [Equation goes here - download the original to see it.] where is rational (not irrational) The infinite series. [Equation goes here - download the original to see it.] Proof For [Equation] the series becomes [Equation goes here - download the original to see it.] Suppose [Equation] The following inequality holds [Equation goes here - download the original to see it.] Similarly, the entire series can be divided into blocks where a similar inequality holds. For instance [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] which shows that it is divergent. 2. [Equation goes here - download the original to see it.] 3. [Equation] [Equation goes here - download the original to see it.] The series [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] is bounded by the limit of this convergent, geometric series; therefore is convergent.
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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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