By James R. Barrante
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Extra resources for Applied Mathematics for Physical Chemistry (2nd Edition)
To emphasize this point, let us assume that the external pressure actually goes up as the volume changes, illustrated by the indicator diagram shown in Fig. 5-2. ) The work done in this case is still the area under a P,,,versus V curve. Measurement of this area, however, is much more difficult than it was when the external pressure was constant. But, bv definition where the symbol the limits of is read "the integral from Vl to V,," and Vl and Vz are called (5-8) The integral in Equation (5-7) is called a definite integral, because it has a fixed value in the interval between VI and V,.
2 4 8 1 6 1 1 1 1 1 (b) - + - + - + - + - + . . 4 12 36 108 324 (i) 2! 3! 4! 24 .. G) 1 1 I+-+-+ 3 42 -531+ - +641. . 2. Using the ratio test, determine whether the following series are convergent or divergent: (a) -1+ - 1+ - +1- + -1+ . 1. (c) -21+ -32+ -43+ - +54 - +65 . . 22 23 32 33 (b) 3 + - + - + 2 3 Figure 7-1 Graph of first five terms of the Fourier series expansion of the function f (x) = 3x, along with graph of the actual function (dashed line). SUGGESTED READING 1. , Upper Saddle River, NJ, 1995.
And that we let S, be the sum of the first n terms. That is, always diverges. (4) The series s , = ~ ~ + ~ ~ + ~ ~ + . . + (7-1) ~ . If, as n approaches infinity, the sum S, approaches a definite, finite value, then the series is said to be convergent. On the other hand, if, as n approaches infinity, S, does not have a definite, finite value but increases without limit, then the series is said to where a > 0,always converges 96 Chapter 7 If each term in an unknown series is less than or equal to a corresponding term in a comparison series that is known to converge, then the unknown series converges.
Applied Mathematics for Physical Chemistry (2nd Edition) by James R. Barrante