Carefully
with infinities
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In the beginning/preface an example of how a finite object can be represented as an infinite number of parts is shown. A whole apple was divided in half, then one half in half again, then one quarter - in half again ...
That is, the integer 1 (apple) was represented as a sum (of pieces)
Note, the number of terms in the sum is infinite (!)
Here is another illustration of how the sum of an infinite number of (decreasing) terms can result in a finite number.
Let's draw several segments one after another, each of which is half the size of the previous one.
Then let's draw square (of corresponding size) on each segment.
Now let's draw a straight line (and this will be a straight line!) through the upper right corners of the squares.
Look carefully and think: this line will intersect the horizontal base of all the segments just there (at that point) where the sum of an infinite number of infinitely decreasing segments converges (!)
It is easy to prove that it is a straight line passes through the upper-right vertices of the squares. Let's write the slope/tangent of any of the segments between the two nearest squares vertices:
That is, the slope/tangent is a constant value for all (pairs of) squares. No matter where they are located.
______________
By the way, another interesting property: if you select any (one) segment, then the (infinite) sum of all segments to the right of it will be equal to ... the length of the selected segment.
____________
However, the infinite sum of decreasing segments (numbers) is not always finite ๐.
However, the infinite sum of decreasing segments (numbers) is not always finite ๐.
For example, the sum of such decreasing terms:
increases to infinity as the number of terms increases.
However, if the sign of the terms alternates (plus to minus and vice versa), then such an infinite sum
finite number (natural logarithm of 2).
ย Hereย
And the sum of the squares of these terms isย
also finite number.
Each of similar (infinite) sums requires a special study๐.
If the terms increase
If the terms increase
If not every sum (with an infinite number) of decreasing terms results in finite number, then what can we say about sums where the terms themselves increase! It is obvious that
1+2+3+โฆ+ n +โฆ
grows up to infinity.
However, it is interesting to look at similar sums with alternating signs in front of the terms. For example, let's try to find the sum S, where
It is not so easy to find result ๐.
Let's try to group the terms:
(1-2) + (3-4) + (5-6)+โฆ = -1 -1 -1โฆ That is, we get S= -โ (negative infinite number).
What if it groups differently?
1+(-2+3)+(-4+5)+โฆ = 1+1+1โฆ That is, we get S = +โ (positive infinity).
And what if to group like this:
Note that at the end of the expression in brackets is ... the infinite sum that we are looking for (S). Thus, it is ok to write:
And that means 4S = 1. And finally: S=1/4 ... Wow! ๐ฎ
To summarize: three different approaches to group the same terms lead to three different results
๐ค๐ฎ๐ง.
It turns out that the sum changes from the rearrangement of the terms?
Yeah! Infinity must be handled with care!
Of course, it's a pity that only decreasing series are summed up (and even then not all of them). Somehow this is strange ...
Or is it possible to come up with something? ๐ค
...and wow! scientists came up with it!
Peculiar and very interesting summation methods were found.
๐
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