Math Is Fun Forum

Another counting system without null:
1-1
2-11
3-111
4-1111
ect.

If you use it very fraquently, you can write in the beginning "S(x) stands for..." and then use it.
I've seen a notebook with properties of the arctangens function, written 'A' for short.

What is "set variable"?

For the series to be convergent, is't nessesary x to be in the unary complex circle: Abs(x)<1.

Whatch this:
http://youtube.com/watch?v=b7fdXLevYvI&

That's not true. You can write it as you like. The difference is only in the domain for n-s. For example, if you're specifying the Fibonnaci sequence, you may write:

or:

it's

It's the only solution. Look at a plot of the above function to see it's strictly increasing for x>1 and x<-1.

And here is the plot - all points lie on a same plane!

You're right, but it's tricky that no complex numbers will occour in the nested radicals!
If you look at the first convergent:

,
you'll see that under the radical there is preciesly positive number, so we'll not have complex numbers at this stage.
But this happens with the second convergent too and with the others... - the last term always eliminates the root-of-negative thing!
So, even paradoxal at first sight, we can't even speak of complex numbers inside the radicals!
Or, in other words, every "sub-radical" is real, and it's bigger enough not to give negative answer when we substract its negative-fellow.

I posted this, because there are identities, extracted by the exactly same way:
A simular nested radical is discovered by Ramanujan:

And this is true! But the last term here is infinity too.
Why this method works in some cases, in others - not?
And indeed, infinity is weird, so this may be true.
I'm maybe not sure the result will be one, but I'm pretty sure that this is not a divergent process!

Calculating on a mental abacus? Cool!

Exaactly!