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)^nx^n/n!
- ((2*n+4)/(5*n+7))^n*x^n
-
(n-2^(3*n))/(n+1)^(1/3)
- (-1)^(2n+1)*x^(2n+1)/(2n+1)!
-
Identical expressions
- (- one)^nx^n/n!
- ( minus 1) to the power of nx to the power of n divide by n!
- ( minus one) to the power of nx to the power of n divide by n!
- (-1)nxn/n!
- -1nxn/n!
- -1^nx^n/n!
- (-1)^nx^n divide by n!
-
Similar expressions
- (1)^nx^n/n!
Sum of series (-1)^nx^n/n!
The solution
You have entered
[src]
oo ____ \ ` \ n n \ (-1) *x / -------- / n! /___, n = 10
$$\sum_{n=10}^{\infty} \frac{\left(-1\right)^{n} x^{n}}{n!}$$
Sum(((-1)^n*x^n)/factorial(n), (n, 10, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n} x^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = -1$$
then
$$R = - \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R = -\infty$$
$$\frac{\left(-1\right)^{n} x^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = -1$$
then
$$R = - \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R = -\infty$$
The answer
[src]
/ 2 4 6 8 9 7 5 3 -x\
10 |-3628800 - 1814400*x - 151200*x - 5040*x - 90*x + 10*x + 720*x + 30240*x + 604800*x + 3628800*x 3628800*e |
x *|------------------------------------------------------------------------------------------------------- + -----------|
| 10 10 |
\ x x /
---------------------------------------------------------------------------------------------------------------------------
3628800
$$\frac{x^{10} \left(\frac{10 x^{9} - 90 x^{8} + 720 x^{7} - 5040 x^{6} + 30240 x^{5} - 151200 x^{4} + 604800 x^{3} - 1814400 x^{2} + 3628800 x - 3628800}{x^{10}} + \frac{3628800 e^{- x}}{x^{10}}\right)}{3628800}$$
x^10*((-3628800 - 1814400*x^2 - 151200*x^4 - 5040*x^6 - 90*x^8 + 10*x^9 + 720*x^7 + 30240*x^5 + 604800*x^3 + 3628800*x)/x^10 + 3628800*exp(-x)/x^10)/3628800
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