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/(2^n)
-
1/(3n-2)*(3n+1)
-
(-1)^n*sqrt(n)/(n+100)
-
ln(1-1/n^2)
-
Identical expressions
- e^(two *n+ one)/n!
- e to the power of (2 multiply by n plus 1) divide by n!
- e to the power of (two multiply by n plus one) divide by n!
- e(2*n+1)/n!
- e2*n+1/n!
- e^(2n+1)/n!
- e(2n+1)/n!
- e2n+1/n!
- e^2n+1/n!
- e^(2*n+1) divide by n!
-
Similar expressions
- e^(2*n-1)/n!
Sum of series e^(2*n+1)/n!
The solution
You have entered
[src]
oo ____ \ ` \ 2*n + 1 \ E / -------- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{e^{2 n + 1}}{n!}$$
Sum(E^(2*n + 1)/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{e^{2 n + 1}}{n!}$$
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
$$a_{n} = \frac{e^{2 n + 1}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(e^{- 2 n - 3} e^{2 n + 1} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$\frac{e^{2 n + 1}}{n!}$$
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
$$a_{n} = \frac{e^{2 n + 1}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(e^{- 2 n - 3} e^{2 n + 1} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
/ 2\ | -2 -2 + e | 3 \- e + e /*e
$$\left(- \frac{1}{e^{2}} + e^{-2 + e^{2}}\right) e^{3}$$
(-exp(-2) + exp(-2 + exp(2)))*exp(3)
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