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For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
Ethanol fuel mixtures have "E" numbers which describe the percentage of ethanol fuel in the mixture by volume, for example, E85 is 85% anhydrous ethanol and 15% gasoline. Low-ethanol blends are typically from E5 to E25, although internationally the most common use of the term refers to the E10 blend. Blends of E10 or less are used in more than ...
The Stoxx 600 index, the region’s benchmark, was trading 0.3% down on the day by 5.53 a.m. ET, having lost 2.2% the day before. London’s FTSE 100 edged 0.3% lower by the same time.
COLOGNE, Germany (AP) — Two Amur tiger cubs had their first public outing Thursday at Cologne Zoo in Germany, one of several zoos that have sought to help keep up the numbers of the rare big ...
Consumer prices rose 2.9% in July compared to a year ago, cooling slightly from the previous month and extending a monthslong slowdown of price increases. Price increases have cooled significantly ...
In this example, the ratio of adjacent terms in the blue sequence converges to L=1/2. We choose r = (L+1)/2 = 3/4. Then the blue sequence is dominated by the red sequence r k for all n ≥ 2. The red sequence converges, so the blue sequence does as well. Below is a proof of the validity of the original ratio test.
Date of birth. Benefits paid each month on: 1st – 10th of the month. The second Wednesday. 11th – 20th of the month. The third Wednesday. 21st – 31st of the month. The fourth Wednesday ...