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Positive and negative infinity
We cannot really imagine what infinity is. This applies both to infinitely large (just think of the universe) and infinitely small. In mathematics you make a difference between positive infinity and negative infinity. That's even more incomprehensible.
Is that just hocus pokus?
Positive infinity
For calculating the number e we use the formula
and try to find the correct answer. We start with simple numbers
Eventually we find the correct answer. That is the transcendental number e = 2.7182….
Negative infinity
In the same way we solve the formula for negative infinity, and write
Once more we start with simple numbers. We begin with n = –1, and get
⁻¹ = 0⁻¹ = 1 ∕ 0 = ﹖.gif){kind=small}
That is undefined. That doesn't help much. So we take n = –2, and see
That is better. Now we continue with n = –3, and find
That is a little less. So we take n = –4, to find out what happens next
The decrease continues. With n = –5, we get
With n = –6, we see
Now we take n = –10, and just write
A bigger step with n = –256 shows
With n = –1024 you see
That comes quite close to what we previously have calculated. So we try n = –4096 and get
This answer is correct to 3 decimals. So here it makes no difference whether we calculate with negative infinity or positive infinity.