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:
-
((-1)^(n+1))/n
- (-1)^n*((x+2)/3)^n*arctg((n+1)/(n^2+4))
-
(-1)^n*10*2^n/(n!)
-
cos(i*n)/n^2
-
Identical expressions
- factorial(n+ one)/ ten ^n
- factorial(n plus 1) divide by 10 to the power of n
- factorial(n plus one) divide by ten to the power of n
- factorial(n+1)/10n
- factorialn+1/10n
- factorialn+1/10^n
- factorial(n+1) divide by 10^n
-
Similar expressions
- factorial(n-1)/10^n
Sum of series factorial(n+1)/10^n
The solution
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[src]
oo ____ \ ` \ (n + 1)! \ -------- / n / 10 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(n + 1\right)!}{10^{n}}$$
Sum(factorial(n + 1)/10^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(n + 1\right)!}{10^{n}}$$
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
$$a_{n} = \left(n + 1\right)!$$
and
$$x_{0} = -10$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-10 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{\left(n + 1\right)!}{10^{n}}$$
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
$$a_{n} = \left(n + 1\right)!$$
and
$$x_{0} = -10$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-10 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{\left(n + 2\right)!}}\right|\right)$$
Let's take the limit
we find
False
$$R = 0$$
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ -n / 10 *(1 + n)! /__, n = 1
$$\sum_{n=1}^{\infty} 10^{- n} \left(n + 1\right)!$$
Sum(10^(-n)*factorial(1 + n), (n, 1, oo))
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