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
-
sin2n
-
lnn/n
- 2(x-6)^(4n)/(n(16)^n)
-
Identical expressions
- cos(n* zero . one)/n!
- co sinus of e of (n multiply by 0.1) divide by n!
- co sinus of e of (n multiply by zero . one) divide by n!
- cos(n0.1)/n!
- cosn0.1/n!
- cos(n*0.1) divide by n!
-
Similar expressions
- (cos(n)*0.1)/n!
- (cos(n)*0.1)/(n!)
Sum of series cos(n*0.1)/n!
The solution
You have entered
[src]
oo ____ \ ` \ /n \ \ cos|--| ) \10/ / ------- / n! /___, n = 0
$$\sum_{n=0}^{\infty} \frac{\cos{\left(\frac{n}{10} \right)}}{n!}$$
Sum(cos(n/10)/factorial(n), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(\frac{n}{10} \right)}}{n!}$$
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
$$a_{n} = \frac{\cos{\left(\frac{n}{10} \right)}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\cos{\left(\frac{n}{10} \right)} \left(n + 1\right)!}{\cos{\left(\frac{n}{10} + \frac{1}{10} \right)} n!}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\cos{\left(\frac{n}{10} \right)} \left(n + 1\right)!}{\cos{\left(\frac{n}{10} + \frac{1}{10} \right)} n!}}\right|$$
$$\frac{\cos{\left(\frac{n}{10} \right)}}{n!}$$
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
$$a_{n} = \frac{\cos{\left(\frac{n}{10} \right)}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\cos{\left(\frac{n}{10} \right)} \left(n + 1\right)!}{\cos{\left(\frac{n}{10} + \frac{1}{10} \right)} n!}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\cos{\left(\frac{n}{10} \right)} \left(n + 1\right)!}{\cos{\left(\frac{n}{10} + \frac{1}{10} \right)} n!}}\right|$$
False
The rate of convergence of the power series
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