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:
- nx^n/(8n-9)*3^n
-
sqrt(n+1)-sqrt(n)/sqrt(n^2+n)
-
cos(i*n)/2^n
-
(3*x^2-2*x+3)/factorial(x+1)
-
Identical expressions
- cos(i*n)/ two ^n
- co sinus of e of (i multiply by n) divide by 2 to the power of n
- co sinus of e of (i multiply by n) divide by two to the power of n
- cos(i*n)/2n
- cosi*n/2n
- cos(in)/2^n
- cos(in)/2n
- cosin/2n
- cosin/2^n
- cos(i*n) divide by 2^n
Sum of series cos(i*n)/2^n
The solution
You have entered
[src]
oo ____ \ ` \ cos(I*n) \ -------- / n / 2 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\cos{\left(i n \right)}}{2^{n}}$$
Sum(cos(i*n)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(i n \right)}}{2^{n}}$$
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
$$a_{n} = \cosh{\left(n \right)}$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{\cosh{\left(n \right)}}{\cosh{\left(n + 1 \right)}}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{\cos{\left(i n \right)}}{2^{n}}$$
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
$$a_{n} = \cosh{\left(n \right)}$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{\cosh{\left(n \right)}}{\cosh{\left(n + 1 \right)}}\right)\right)$$
Let's take the limit
we find
False
False
$$R = 0$$
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ -n / 2 *cosh(n) /__, n = 1
$$\sum_{n=1}^{\infty} 2^{- n} \cosh{\left(n \right)}$$
Sum(2^(-n)*cosh(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