Mister Exam
Other calculators
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How to use it?
- Sum of series:
- (2^n+3^n)/6^n
- 1/n!
- x^n/2^n
- 1/(3n-2)*(3n+1)
-
Identical expressions
- factorial(n)* three ^n/n^n
- factorial(n) multiply by 3 to the power of n divide by n to the power of n
- factorial(n) multiply by three to the power of n divide by n to the power of n
- factorial(n)*3n/nn
- factorialn*3n/nn
- factorial(n)3^n/n^n
- factorial(n)3n/nn
- factorialn3n/nn
- factorialn3^n/n^n
- factorial(n)*3^n divide by n^n
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Expressions with functions
- factorial
- factorial(x-n)
- factorial(n)^2/factorial(k^n)
- factorial(k)/((factorial(n)*factorial(n+k)))
- factorial(n)/n^n
- factorial(n)*factorial(2*n+2)/factorial(3*n)
Sum of series factorial(n)*3^n/n^n
The solution
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[src]
oo ____ \ ` \ n \ n!*3 ) ----- / n / n /___, n = 1
$$\sum_{n=1}^{\infty} \frac{3^{n} n!}{n^{n}}$$
Sum((factorial(n)*3^n)/n^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3^{n} n!}{n^{n}}$$
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
$$a_{n} = n^{- n} n!$$
and
$$x_{0} = -3$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(n^{- n} \left(n + 1\right)^{n + 1} \left|{\frac{n!}{\left(n + 1\right)!}}\right|\right)\right)$$
Let's take the limit
we find
$$\frac{3^{n} n!}{n^{n}}$$
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
$$a_{n} = n^{- n} n!$$
and
$$x_{0} = -3$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(n^{- n} \left(n + 1\right)^{n + 1} \left|{\frac{n!}{\left(n + 1\right)!}}\right|\right)\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ n -n / 3 *n *n! /__, n = 1
$$\sum_{n=1}^{\infty} 3^{n} n^{- n} n!$$
Sum(3^n*n^(-n)*factorial(n), (n, 1, oo))
The graph