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The following outline is provided as an overview of and topical guide to the September 11 attacks and their consequences: September 11 attacks – four coordinated suicide attacks upon the United States in New York City and the Washington, D.C., area on September 11, 2001.
S n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail) He used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.141 592 653 59. He also improved the formula based on arctan(1) by including a correction:
For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form f ( x ) = a · b x appear as straight lines with slope equal to the logarithm of b .
In other words, column j of A expresses the image of v j in terms of the basis vectors w I of W; thus this relation uniquely determines the entries of the matrix A. The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices . [ 61 ]
For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields [34] + + + = + + + = + + + = (+ + +), which is times the original series, so it would have a sum of half of the natural logarithm of 2. By ...
The convergents of the continued fraction for φ are ratios of successive Fibonacci numbers: φ n = F n+1 / F n is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φ n+1 = 1 + 1 / φ n. [32] The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities ...
The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition .
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