When n->; When ∞, the limit of (1+ 1/n) n 。
Note: x y stands for the y power of x.
With the increase of n, the cardinality is closer to 1, while the exponent tends to infinity. Does the result tend to be 1 or infinite? Actually, it tends to be 2.7 1828 ... If you don't believe me, use a calculator to calculate, and take n = 1, 10 and 1000 respectively. However, because the general calculator can only display about 10 digits, it is necessary to read more.
E is widely used in science and technology. Logarithms based on 10 are generally not used. Taking e as the base can simplify many formulas and is the most "natural" to use, so it is called "natural logarithm".
The e here is a symbol of a number, and we are going to tell the story of e, which makes people a little curious. If it can be said to be a book, this number should be a household name, at least it should be famous. However, when searching for impoverished mind, apart from the well-known numbers 0 and 1, most people can only think of π related to the circle, and surprisingly add the imaginary unit i=√- 1. Where is this e?
In high school mathematics, everyone has learned the concept of logarithm and can use logarithm. Logarithms in textbooks are all based on 10, which is called ordinary logarithm. Textbooks also briefly mention that there is a logarithm of irrational number E = 2.7 1828 ... this is the so-called natural logarithm. Is it more natural to use such a strange number as the base in decimal system than to use 10 as the base? What is even more curious is, what is the story of such a strange number?
This should start from ancient times. This number was mentioned at least half a century before the invention of calculus, so although it often appears in calculus, it was not born with calculus. So under what circumstances did it appear? One possible explanation is that this figure is related to the calculation of interest.
We all know what compound interest is, that is, interest can be regenerated with the principal. But the sum of principal and interest depends on the interest period. In a year, interest can be calculated once a year, once every six months, once a quarter, once a month or even once a day. Of course, the shorter the interest period, the higher the sum of principal and interest. Some people are curious about this. What will happen if the interest period is shortened indefinitely, such as once a minute, even once a second, or all the time (theoretically)? Will the principal and interest rate increase indefinitely? The answer is no, its value will be stable and close to a limit value, and the number e will appear in the limit value (of course, the number has not been named e at that time). So in today's mathematical language, e can be defined as a limit value, but there was no concept of limit at that time, so the value of e should be observed, not obtained through strict proof.