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)^n
-
3^(n+3)/(4^(n-1)*5^n)
-
sin((n^2+3)/(n^3+2))^2
-
6/(3n-1)(3n+5)
-
Identical expressions
- sin three ^n/3^n
- sinus of 3 to the power of n divide by 3 to the power of n
- sinus of three to the power of n divide by 3 to the power of n
- sin3n/3n
- sin3^n divide by 3^n
Sum of series sin3^n/3^n
The solution
You have entered
[src]
oo ____ \ ` \ n \ sin (3) ) ------- / n / 3 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin^{n}{\left(3 \right)}}{3^{n}}$$
Sum(sin(3)^n/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin^{n}{\left(3 \right)}}{3^{n}}$$
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
$$a_{n} = \sin^{n}{\left(3 \right)}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\sin^{n}{\left(3 \right)} \sin^{- n - 1}{\left(3 \right)}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{\sin^{n}{\left(3 \right)}}{3^{n}}$$
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
$$a_{n} = \sin^{n}{\left(3 \right)}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty}\left(\sin^{n}{\left(3 \right)} \sin^{- n - 1}{\left(3 \right)}\right)\right)$$
Let's take the limit
we find
False
False
$$R = 0$$
The rate of convergence of the power series
The answer
[src]
sin(3) -------------- / sin(3)\ 3*|1 - ------| \ 3 /
$$\frac{\sin{\left(3 \right)}}{3 \left(1 - \frac{\sin{\left(3 \right)}}{3}\right)}$$
sin(3)/(3*(1 - sin(3)/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