Huh.
is really just a special case of
where the constant denominator d is 1.
The very nature of sums requires that the sum be defined when x = 1, which it often is not.
This should help.
I like playing with math and numbers and have always liked the idea of summation formulas. This is what I played with yesterday.
I started with
I found it needed a constant modifier which resulted in
The denominator is the constant d, n1 is the first value of n, ln is the final value of n.
When we can reproduce what others have done before, that is school work.
I would be interested in links to a proof of this or where it can be found elsewhere.
I have not tried it with fractional exponents or with especially large values of n, but it does work with decimals greater than one.
x ( x^{n} – 1) | |
∑ x^{n} = | ____________ |
(x – 1) |
Salt can be very therapeutic, but a trip to the beach is usually quite expensive.
According to WorldBook Encyclopedia, the ratio of salt to water in the ocean averages about 35 to 965, which is about 1:28.
The salinity of Salt Lake is about 7 times that of the ocean, which would make the salt to water ratio about 1:4. The salinity of the Dead Sea is about 9 times the salinity of the ocean, which would make that ratio about 1:3.
That is a lot of salt to drain from a bathtub every day.
To know how much salt to use, it helps to know how many gallons are in a square foot and knowing the specific gravity (the ratio of the density to the density of water) of salt, also known as sodium chloride.
I wonder what grows in the Dead Sea that brine shrimp eat?
Most organisms decay in water. I soaked goathead seeds in a covered container of water for a full year. The water developed algae and became green, but at the end of the year, the seeds were as strong as ever. They did not decay at all.
I have collected every seed I could find that I have stopped on, so I had plenty to experiment with. I used readily available household cleaners including dish detergent, vinegar and salt water.
It looks like the seeds can tolerate salt water, for a short while anyway.
Noxious but robust weed seeds soaked in salt water for 2 weeks
The most impressive results came from the seeds soaked in straight bleach.
Weed seeds decaying after two weeks soaking in straight bleach
The picture does not do justice to the beauty of the decay.
Few seeds are easy to drop into a jar of bleach, but the same result could likely be achieved by soaking a blanket or towel in bleach and covering it with plastic to keep it from drying out so fast from evaporation.
My mother swore uphill and down that garlic was healthy, partly because it thins the blood. She did insist, though, that it had to be fresh and raw, not cooked or processed.
With all the summer-time bugs, it occurred to me to take another look at vampire legends that garlic repels blood suckers. Even the thought of swallowing raw garlic whole I found a little daunting, so I made a variation on a pill.
I mixed 1 teaspoon of whole wheat flour with 1/4 to 1/2 teaspoon water to make a tiny ball of dough. Then I took 1 small clove of fresh garlic (can use 1/2 of a medium-size clove or 1/4 of a large clove) and wrapped it in the dough ball. Swallowed it whole and washed it down with a glass of water.
The repellence appears to be quite effective.