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:
-
(2/5)^n
- (4x)^(2n)
-
3^n/n^2
-
1/5^n
-
Identical expressions
- n^ three *e^(two *n+ one)/factorial(n)
- n cubed multiply by e to the power of (2 multiply by n plus 1) divide by factorial(n)
- n to the power of three multiply by e to the power of (two multiply by n plus one) divide by factorial(n)
- n3*e(2*n+1)/factorial(n)
- n3*e2*n+1/factorialn
- n³*e^(2*n+1)/factorial(n)
- n to the power of 3*e to the power of (2*n+1)/factorial(n)
- n^3e^(2n+1)/factorial(n)
- n3e(2n+1)/factorial(n)
- n3e2n+1/factorialn
- n^3e^2n+1/factorialn
- n^3*e^(2*n+1) divide by factorial(n)
-
Similar expressions
- n^3*e^(2*n-1)/factorial(n)
Sum of series n^3*e^(2*n+1)/factorial(n)
The solution
You have entered
[src]
oo ____ \ ` \ 3 2*n + 1 \ n *E / ----------- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{e^{2 n + 1} n^{3}}{n!}$$
Sum((n^3*E^(2*n + 1))/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{e^{2 n + 1} n^{3}}{n!}$$
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
$$a_{n} = \frac{n^{3} e^{2 n + 1}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n^{3} e^{- 2 n - 3} e^{2 n + 1} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{\left(n + 1\right)^{3}}\right)$$
Let's take the limit
we find
$$\frac{e^{2 n + 1} n^{3}}{n!}$$
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
$$a_{n} = \frac{n^{3} e^{2 n + 1}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{n^{3} e^{- 2 n - 3} e^{2 n + 1} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{\left(n + 1\right)^{3}}\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
/ 2\ / 2 4\ 3 \e / \1 + 3*e + e /*e *e
$$\left(1 + 3 e^{2} + e^{4}\right) e^{3} e^{e^{2}}$$
(1 + 3*exp(2) + exp(4))*exp(3)*exp(exp(2))
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