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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
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:
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.
For sale or trade, 3" & 3 1/2" steel shotgun ammo, 12 gage. Don't know the velocity but it is #2 shot. Price is $15 per box FTF in the Lebanon/Albany area. I WILL NOT ship. Trade interests are WWII US Military blades and maybe factory ammo especially 12 gage mini shells. I'm like everyone...
For sale or trade is the following shotgun ammo from an estate. Friend of a friend and it has been stored inside the home.
Federal 3 1/2" #2 steel shot--6 full boxes and one with 22 rounds
Federal 3" #2 steel shot--1 full box
Federal 2 3/4" #2 steel shot--1 full box
Remington 3" #3 steel...
decided to part ways because I’m looking to get into a kel-tec ksg. It’s barely been fired through my year of ownership, mostly stored away (looks brand new inside & out) you can take it apart and look for yourself. It would’ve made a great hunting shotgun i just never got around to it.
decided to part ways because I’m looking to get into a kel-tec ksg. It’s barely been fired through my year of ownership you can take it apart and look for yourself. It would’ve made a great hunting shotgun i just never got around to it.
It’s a stoeger parent company benelli (feels great)
I've got 8 boxes of #4 buck for sale great for coyote or home defense if your gun and shoulder can handle 3 1/2" loads.
Asking $120.00 for the lot. can meet up in Sandy or Gresham.
Not looking for trades but feel free to offer.
Just one box of 3 1/2 " BBs and I no longer have a 3 1/2" chambered shotgun.
Maybe someone can finish out goose season with them.
$12. I'd trade for 2 boxes of 12 or 20 gauge #8, 9, or 7 1/2s.
Prefer to meet at Cabelas just off I-5 at the Tualatin exit or Dicks on Boone’s Ferry or Fisherman's...
870 Barrel. Not sure where measurement is taken but there's a pic of the end of the tape. Smooth Bore, open to trades for ammo or cash offers. Open to guns as well with cash I can throw on top. Barrel has very low shot count, Pickup in Beaverton or can ship for 30$.
I have 10 boxes of 12 gauge Winchester buck shot - three and a half inch. These were bought in April and stored in a sealed cans. $15 per box or $140 takes them all.
I thought I'd try to put this up here first. It's a gun I recently inherited and have no use for.
It's a Zabala 10ga SXS 32" barrel, Full & Full fixed chokes, 3 1/2" magnum, made in Spain.
It's in decent condition but not great, barrels look clean and shiny, stock is solid but has some dings...
FUNDS TO BE USED FOR DONATION TO
Maybe you are interested in this.
I have a partial box of Western Super X Magnum shotgun shells, 3 1/2", 2 oz , 2 shot. 22 rounds in the box. I want $10.
I also have around 10 loose 20 gauge, 2 3/4" 1oz 6 shot shells I can include...
Mossberg 835 Ultimag 12ga. shotgun, 28" barrel. Takes 2 3/4" to 3 1/2". Bought it for turkey, but never got around to going. It has taken its share of upland and geese. green synthetic stock with black matte finish, ported Barrel. Gun is in good condition, normal wear. Asking $200. Would...