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3-1/2
In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
∑
n
=
1
m
n
(
−
1
)
n
−
1
.
{\displaystyle \sum _{n=1}^{m}n(-1)^{n-1}.}
The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
1
−
2
+
3
−
4
+
⋯
=
1
4
.
{\displaystyle 1-2+3-4+\cdots ={\frac {1}{4}}.}
A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts. Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required.
The series 1 − 2 + 3 − 4 + ... is closely related to Grandi's series 1 − 1 + 1 − 1 + .... Euler treated these two as special cases of the more general sequence 1 − 2n + 3n − 4n + ..., where n = 1 and n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
I am helping a buddy to sell off some of his guns.
He recently purchased another Browning and so this needs a new home.
He bought this used about 6 years ago and has shot a couple hundred round though it himself.
I do not know much about shotguns, so I tried to take as many pics as possible to...
I have one full box + 3 loose shells of Kent Impact Tungsten Matrix 12ga 3-1/2" 2oz #3 shot.
I paid $70 per box, but will sell the full box +3 loose shells for $60 $50 obo within a reasonable distance of the Beaverton area or will ship for + $10.
I have a partial box of 9 shells of Federal Premium Turkey Shot 12ga 3-1/2" 2-1/4oz #6 Shot.
Reduced price because I misplaced one shell.
$20 $15 obo within a reasonable distance of the Beaverton area or will ship for + $10.
I have a partial box of 9 shells of Federal Premium Turkey Shot 12ga 3-1/2" 2-1/4oz #6 Shot.
Reduced price because I misplaced one shell.
$20 obo within a reasonable distance of the Beaverton area or will ship for + $10.
Benelli Super Vinci 12 Ga 3-1/2" mag
28" barrel, used 2-seasons and in great condition, comes in original Benelli case and original set of chokes
$1000.00 Cash, not interested in trades
After 1st of the year I will consider shipping, upon cleared payment, to buyers FFL at buyers expense
3-1/2 IN Shotshell 1-3/8oz 2shot Winchester super X, 75 rounds, $80.00 each (2 available)
Need the money more than the ammo one box has been opened but no ammo has been used.