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This has a finite limit as t goes to infinity, namely π /2. Similarly, the integral from 1/3 to 1 allows a Riemann sum as well, coincidentally again producing π /6. Replacing 1/3 by an arbitrary positive value s (with s < 1) is equally safe, giving π/2 − 2 arctan(√ s). This, too, has a finite limit as s goes to zero, namely π /2 ...
The music video for the song premiered on the MySpace main page January 16, 2009 [3] and was subsequently released on MTV, MTVU, VH1, Fuse, Music Choice and YouTube. [4] [5] [6] It found success on the weekly VH1 Top 20 Video Countdown, charting over five months straight between January and May, peaking at #5.
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
Larry Flick from Billboard described the song as "a jumpy, funk-lined jeep anthem that allows Coolio plenty of room to work up a fun, lyrical sweat."He added, "The sample-happy groove provides a wigglin' good time, riding primarily on a prominent snippet of the early '80s 12-incher "Wikka Wrap" by the Evasions.
1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. In mathematics, the infinite series 1 2 + 1 4 + 1 8 + 1 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation ...
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Illustration of the poem from the 1901 Book of Nursery Rhymes. This is one of many counting-out rhymes. It was first recorded in Mother Goose's Melody around 1765. Like most versions until the late 19th century, it had only the first stanza and dealt with a hare, not a fish, with the words:
The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.