Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
exp(n)
- sin
- f(x)
- f(xi)
- Limit of the function:
-
exp(n)
-
Identical expressions
- exp(n)
- exponent of (n)
- expn
-
Similar expressions
- ln(n/10)/(exp(n/10)+1)
- exp^(n^2)
- exp(n*(-3)/2)
- ln(n/10)/(exp(n/10)-1)
- exp(nx)
Sum of series exp(n)
The solution
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oo ___ \ ` \ n / e /__, n = 1
$$\sum_{n=1}^{\infty} e^{n}$$
Sum(exp(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$e^{n}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = 1$$
and
$$x_{0} = - e$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
$$e^{n}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = 1$$
and
$$x_{0} = - e$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(- e + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False
The rate of convergence of the power series
Numerical answer
The series diverges
The graph
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n