Mister Exam
Other calculators
- 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:
-
factorial(n+1)/n^n
-
(-1)^(n+1)/n
-
log((n+1)/(n+2))
- 1/(k*(k+1)*(k+2))
-
Identical expressions
- factorial(n+ one)/n^n
- factorial(n plus 1) divide by n to the power of n
- factorial(n plus one) divide by n to the power of n
- factorial(n+1)/nn
- factorialn+1/nn
- factorialn+1/n^n
- factorial(n+1) divide by n^n
-
Similar expressions
- factorial(n-1)/n^n
Sum of series factorial(n+1)/n^n
The solution
You have entered
[src]
oo ____ \ ` \ (n + 1)! \ -------- / n / n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{n^{n}}$$
Sum(factorial(n + 1)/n^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(n + 1\right)!}{n^{n}}$$
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
$$a_{n} = n^{- n} \left(n + 1\right)!$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(n^{- n} \left(n + 1\right)^{n + 1} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find
$$\frac{\left(n + 1\right)!}{n^{n}}$$
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
$$a_{n} = n^{- n} \left(n + 1\right)!$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(n^{- n} \left(n + 1\right)^{n + 1} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find
False
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ -n / n *(1 + n)! /__, n = 1
$$\sum_{n=1}^{\infty} n^{- n} \left(n + 1\right)!$$
Sum(n^(-n)*factorial(1 + n), (n, 1, oo))
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