The limit of the sequence
For infinite sequences, we have learned a lot in high school, and some of them have characteristics, and we look at examples.
We can observe that
It will get smaller and smaller, and eventually infinitely close to 0
One positive and one negative, but eventually it will also approach 0 indefinitely
They will get bigger and bigger, and while it's not easy to see, they will eventually approach a constant of 2
And the corresponding two other sequences
Obviously, in the end it does not approach a constant, the former jumps between two numbers, and the latter approaches infinity.
Then the mathematicians came on, and they said, let's define it.
If the sequence approaches a constant, then the sequence is said to be convergent, otherwise it is said to be divergent. And with symbols
In the example above, we can write it like this
It seems to be very clear. But this directly shows that mathematicians are very low-end, so they developed a language specifically for describing limits, but that is really just a mathematician's play, let's just ignore him.
What the? What the? Would you like to see this description of a mathematician? Well, I'll show you the show, I can understand it, I can't understand it and pull it down.
My personal feeling is, what a mess to say. Obviously very simple and understandable things, how complicated to say!
Of course, like the number series cited in this article, the limit is easy to see, and the limit of some number series is not easy to see, and it needs to be calculated a little. For example
It's not so easy to see. This requires some calculations and simple conclusions, and the following conclusions obviously cannot be more obvious.
With these simple and obvious conclusions, the slightly more complex limits are fine.
With the tool for the limits of the sequence, the famous Zeno paradox can be easily cracked:
Let the turtle start 1000 meters in front of the athlete Achilles, race against Achilles, and assume that Achilles is 10 times faster than the turtle. When the race starts, if Achilles runs 1000 meters, set the time t, then the turtle will lead him by 100 meters; when Achilles runs the next 100 meters, the time he uses is t/10, the turtle is still 10 meters ahead of him; when Achilles runs the next 10 meters, the time he uses is t/100, and the turtle is still 1 meter ahead of him... Zeno believes that Achilles will be able to continue to approach the turtle, but it will never be possible to catch up with it.
We can set that Achilles ran to the first turtle point and took t
Achilles ran to the second turtle spot, which took t/10
Achilles ran to the third point, which took t/100
……
Achilles runs to the nth point, which takes t/10^n
So Achilles caught up with the turtle's total time
That is to say, the time for Achilles to catch up with the turtle is limited, although it is an infinite series, but the sum of the sequences is not infinite.