Finite minds determining the nondeterministic infinite things

we see but mathematicians observe
There is a drastic difference between seeing and observing .The difference is,seeing means looking at a shallow manner without diving but observing means diving into it and then telling it.
Mathematicians are best virtual divers .They plunge into the world of abstract mathematics and they atlast come to the conclusion .
Here one more thing that I want to mention it,mathematics subject itself is a fractal,when you dive into it you can see that you are again at the top layer no matter how many layers you go into it .Mathematics is a subject where suprises hidden within suprises.Mathematics possess beauty which has no end to it.
I will explain this with some simple examples
Suppose consider the sum of infinite series 1+1/3+1/6+1/10+1/15+…..
Here you can see what the above numbers they are the reciprocals of nth triangular numbers.
When n=1,n(n+1)/2 gives 1
When n=2 , n(n+1)/2 gives 3
When n=3 , n(n+1)/2 gives 6
When n=4 , n(n+1)/2 gives 10
And so on .
Ok now you know where each denominator came from .
Ok when we see this infinite series we are unable to predict the sum of this infinite series but this can be solved this in simple algebraically.
First algebraically we will solve it
Let s=1+1/3+1/6+1/10+……………….
Now multiply both sides by ½
(½)s=1/2+1/6+1/12+1/20+…………….
Now with the help of simple logic ½ can be written as (1-1/2) and similarly 1/6 can be written as(1/2-1/3) and 1/12 can be written as (1/3-1/4) and so on.
=>(1/2)s=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+…………..
So all the terms will cancel except 1
So (1/2)s=1
=>s=2.
=>1+1/3+1/6+1/10+……………….=2
Hey hey wait one minute now consider
S=1+1/3(part of this infinite series and compute it in the same manner)
=>(1/2)s=(1/2)+(1/6)
=>(1/2)s=(1-1/2)+(1/2-1/3)
=>(1/2)s=1-1/3
=>(1/2)s=2/3
=>s=4/3
So now considering the finite sums like these you may tell that 1 and the last number will also remains .
So which implies you are seeing whereas mathematicians know last term will diminish when we are considering the infinite series since they are observing(How the last term diminishes .Answer is very simple in case of infinite series the last term will be a very big number reciprocal which is nearly zero .)
Ok now consider the another series 1+1/2+1/3+1/4+1/5+………..
Which is nothing but reciprocal of the sum of the natural numbers .by considering the previous sum of series we may tell that sum of the series might be some finite number which is not happening in reality.
We will see why it is
S=1+1/2+1/3+1/4+1/5+……………..
s=1+(1/2)+(1/3+1/4)+91/5+1/6+1/7+1/8)+(1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16)+……………
We are grouphing in terms of 2 power n and also note that
¼<1/3 and 1/8<1/5,1/6,1/7 and 1/16<1/15,1/14,1/13,1/12,1/11,1/10,1/9 and so on …..
>1+(1/2)+(1/4+1/4)+(1/8+1/8+1/8+1/8)+……………….
=1+(1/2) +(1/2) +(1/2) +(1/2)+…………..
So the infinite (1/2) terms quickly diverge to infinity (means they will turns to a huge number) so the subset itself diverges
So the original series 1+1/2+1/3+1/4+1/5+………….. diverges to infinity(very big number ) which is counter intituitive.
Now another example
Find the value of

Which we can’t predict it value since we are seeing but where as mathematicians are observing using simple algebraic operations we can able to find out its value. Actually the value is very interesting and it is called golden ratio(1.608..) which has pretty interesting features. Similarly one more example
Find the value of

Which can also be solved very easily using simple algebraic operations and geometric series help we can see that its value is 3
So We can see that finite mind(mathematicians) can determine the infinite intrinsic beauties.