Introduction to infinite series e:
The celebrated Greek mathematician, engineer and scientist Archimedes introduced the constant Π. Similarly the great Swiss mathematician Leonhard Euler introduced a constant denoted as e. It is called exponential constant. This constant has a great use in calculus, similar to the constant Π in geometry. It is an irrational number equal to 2.71828……
Apart from mathematics, the exponential constant has many application in science, statistics and in other fields. Some of the growth and decay rates are associated with this constant.
There are many interesting features with this constant. One of them is it is the sum of an infinite series called infinite series e. Understanding Infinite Limits is always challenging for me but thanks to all math help websites to help me out.
Definition of Infinite Series E –
Euler defined e as,
e = 1 + $ \frac{1}{1!}$ + $ \frac{1}{2!}$ + $ \frac{1}{3!}$ + ……. $ \frac{1}{n!}$ + …….
The value of e can be proved to lie between 2 and 3. Since
e = 1 + $ \frac{1}{1!}$ + $ \frac{1}{2!}$ + $ \frac{1}{3!}$ + ……. $ \frac{1}{n!}$ + …….
or e > 1 + 1 + positive fractions
Obviously e > 2
The fourth term in the series is $ \frac{1}{3!}$, that is, $ \frac{1}{6}$
It is less than $ \frac{1}{4}$ which is $ \frac{1}{2^2}$
Similarly the next term $ \frac{1}{4!}$ is less than $ \frac{1}{2^3}$
Therefore, e < 1 + 1 + $ \frac{1}{2}$ + $ \frac{1}{2^2}$ + $ \frac{1}{2^3}$ + ……
or, e < 1 + $ \frac{1}{1 - 1/2}$
or, e < 1 + 2
or, e < 3
Hence the value of e lies between 2 and 3
Is this topic Chain Rule Integration hard for you? Watch out for my coming posts.
Infinite Series E – Power Series:
The power expression ex is defined as,
ex = 1 + $ \frac{x}{1!}$ + $ \frac{x^2}{2!}$ + $ \frac{x^3}{3!}$ + …… $ \frac{x^n}{n!}$ + …
Now let us see an interesting result.
Differentiating both sides,
Derivative of ex = 0 + 1 + $ \frac{2x}{2!}$ + $ \frac{3x^2}{3!}$ + …… $ \frac{nx^n-1}{n!}$ + … $ \frac{(n+1)x^n}{(n + 1)!}$
= 1 + $ \frac{x}{1!}$ + $ \frac{x^2}{2!}$ + $ \frac{x^3}{3!}$ + …… $ \frac{x^n}{n!}$ + … = ex
The celebrated Greek mathematician, engineer and scientist Archimedes introduced the constant Π. Similarly the great Swiss mathematician Leonhard Euler introduced a constant denoted as e. It is called exponential constant. This constant has a great use in calculus, similar to the constant Π in geometry. It is an irrational number equal to 2.71828……
Apart from mathematics, the exponential constant has many application in science, statistics and in other fields. Some of the growth and decay rates are associated with this constant.
There are many interesting features with this constant. One of them is it is the sum of an infinite series called infinite series e. Understanding Infinite Limits is always challenging for me but thanks to all math help websites to help me out.
Definition of Infinite Series E –
Euler defined e as,
e = 1 + $ \frac{1}{1!}$ + $ \frac{1}{2!}$ + $ \frac{1}{3!}$ + ……. $ \frac{1}{n!}$ + …….
The value of e can be proved to lie between 2 and 3. Since
e = 1 + $ \frac{1}{1!}$ + $ \frac{1}{2!}$ + $ \frac{1}{3!}$ + ……. $ \frac{1}{n!}$ + …….
or e > 1 + 1 + positive fractions
Obviously e > 2
The fourth term in the series is $ \frac{1}{3!}$, that is, $ \frac{1}{6}$
It is less than $ \frac{1}{4}$ which is $ \frac{1}{2^2}$
Similarly the next term $ \frac{1}{4!}$ is less than $ \frac{1}{2^3}$
Therefore, e < 1 + 1 + $ \frac{1}{2}$ + $ \frac{1}{2^2}$ + $ \frac{1}{2^3}$ + ……
or, e < 1 + $ \frac{1}{1 - 1/2}$
or, e < 1 + 2
or, e < 3
Hence the value of e lies between 2 and 3
Is this topic Chain Rule Integration hard for you? Watch out for my coming posts.
Infinite Series E – Power Series:
The power expression ex is defined as,
ex = 1 + $ \frac{x}{1!}$ + $ \frac{x^2}{2!}$ + $ \frac{x^3}{3!}$ + …… $ \frac{x^n}{n!}$ + …
Now let us see an interesting result.
Differentiating both sides,
Derivative of ex = 0 + 1 + $ \frac{2x}{2!}$ + $ \frac{3x^2}{3!}$ + …… $ \frac{nx^n-1}{n!}$ + … $ \frac{(n+1)x^n}{(n + 1)!}$
= 1 + $ \frac{x}{1!}$ + $ \frac{x^2}{2!}$ + $ \frac{x^3}{3!}$ + …… $ \frac{x^n}{n!}$ + … = ex
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