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:
- x^n/n
-
1/n^6
-
1/n^n
-
(n-1)/n!
-
Identical expressions
- - one ^n((n+ one)^n+ one)/(n+ one)!
- minus 1 to the power of n((n plus 1) to the power of n plus 1) divide by (n plus 1)!
- minus one to the power of n((n plus one) to the power of n plus one) divide by (n plus one)!
- -1n((n+1)n+1)/(n+1)!
- -1nn+1n+1/n+1!
- -1^nn+1^n+1/n+1!
- -1^n((n+1)^n+1) divide by (n+1)!
-
Similar expressions
- -1^n((n+1)^n-1)/(n+1)!
- -1^n((n-1)^n+1)/(n+1)!
- 1^n((n+1)^n+1)/(n+1)!
- -1^n((n+1)^n+1)/(n-1)!
Sum of series -1^n((n+1)^n+1)/(n+1)!
The solution
You have entered
[src]
oo ____ \ ` \ n / n \ \ -1 *\(n + 1) + 1/ / ------------------ / (n + 1)! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{- 1^{n} \left(\left(n + 1\right)^{n} + 1\right)}{\left(n + 1\right)!}$$
Sum(((-1^n)*((n + 1)^n + 1))/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{- 1^{n} \left(\left(n + 1\right)^{n} + 1\right)}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{- \left(n + 1\right)^{n} - 1}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{n} + 1\right) \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|}{\left(n + 2\right)^{n + 1} + 1}\right)$$
Let's take the limit
we find
$$\frac{- 1^{n} \left(\left(n + 1\right)^{n} + 1\right)}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{- \left(n + 1\right)^{n} - 1}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{n} + 1\right) \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|}{\left(n + 2\right)^{n + 1} + 1}\right)$$
Let's take the limit
we find
False
False
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ n \ -1 - (1 + n) / ------------- / (1 + n)! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{- \left(n + 1\right)^{n} - 1}{\left(n + 1\right)!}$$
Sum((-1 - (1 + 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