Right before They speak on Converg n Diverg Series

We know on studying series, we have some tight rules. One of it is the convergent of a define series. a Series with equal to S have to prove its convergent or not. Or its should convergent on its tail.

But right before it, we don't use it convergent to determine the sum of series. Its a nice sample below n how mathematicians confuse before.

Define a series below

1 - 1 + 1 - 1 + 1 - ...

So, when we first meet this situation, we will do an association that the series will become like this.

(1 - 1) + (1 - 1) + ...

Since for each (1-1) equal to zero, we know that the whole series is equal to Zero too..
But now think about doing same with another different point of views, and we get the new series below

1 + (-1 + 1) + (-1 + 1) + ...

Since for each (-1 + 1) is equal to zero, we will find it left a one on that series. Then we conclude that the whole series is equal to 1.

That make a paradox. How could an identical series have 2 solution for S, that S is sums of series? And we know that the procedure to get the 2 of solution is comfirmed as legal. And you will get more confuse (before the convergent spoked) with this method. We use whole series as a symbol we called S, as below.

S = 1 - 1 + 1 - 1 + ...

and We do some distribution n association below

S = 1 - (1 - 1 + 1 - 1 + ... )

since we know that S is the series itself,

S = 1 - S

then we add an S to both side and we get

2S = 1

So we conclude that S, that is whole series itself, is equal to an half. The solution will lead us to 0 = 1 = 1/2, that is insane, so the consensus make the convergent of the series as a factor to satisfy before we find the solution.

Early Question on Negative Number

In 17th century, the ratio of positive integers and negative integers are as follows:

-n / n = -1 = n /-n

Above equation seems normal. But we know -n is less than n, and give an equal sign in the expression on the right and the left, ie, dividing the smaller number with larger numbers is equal to dividing the larger number with smaller numbers, which of course we would think it impossible. This is called the paradox of Arnauld.

To accomplish this, we can not assume-n (negative integer) as usual quantity, but as the quantity with a different direction. It looks very natural given the integer applications in everyday life, such as stepping forward and stepping backward, or the money we spend and what we get. With this, we no longer consider the comparison '-n / n' and 'n /-n' as the ratio of quantity only. When the ratio of expanded notation for negative numbers as in the positive numbers, terminology b/c can be understood as the number needed to be multiplied by c to get b. Multiplication number k.-j with negative numbers can be understood as a multiplication with the absolute value but pay attention to the direction of the inverse multiplication k.j.

Why math love 'letter' than 'numbers'

Okey, I'm sure, you often see minds readers. We will show how algebra provides an easy to use letters when compared with using numbers directly. Let us remember the words of forecaster who want to read our minds.

1. Think of a number between 1 and 10
2. Add to 3
3. Multiply the two, numbers that you get
4. Add the number you're thinking to the result before
5. Devide the last by 3
6. Take the last number that you get with what number you're thinking
7. The last number you get is 2

The question that we often think about is how the mind readers to find out our final answer without knowing the first number we think, is not it?

I know what happens at each step as if I chose to use letters rather than a specific number. For example I use x and you use the 7 in the first step. The following process, namely the steps above, would explain why x becomes important.

You Me
7 x

10 x + 3 (unknown number plus 3)

20 2(x + 3) = 2x + 6 ('multiply two')

27 (2x + 6) + x = 3x + 6 ('add to the number of unknown')

9 (3x + 6) / 3 = x + 2 ('divide by 3')

2 2 ('take an unknown number of the last number')


Of the two processes we have the same rid first value that we think so as to obtain results that course. By using letters, its more easily to manage the process as a mind reader, also understand why we get the same results, ie because the numbers that we do not know (forecasters), or number of the selected assistant forecasters ('example above 7') we discard the process in a way that carefully so assistant forecasters do not sense it.