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/((4n-3)(4n+1))
-
(3x^2-2x+3)(x+1)^x
-
sqrt(1-(2*n-1)/(2*n))
-
sin(1/(3^n))
-
Identical expressions
- sin(one /(three ^n))
- sinus of (1 divide by (3 to the power of n))
- sinus of (one divide by (three to the power of n))
- sin(1/(3n))
- sin1/3n
- sin1/3^n
- sin(1 divide by (3^n))
Sum of series sin(1/(3^n))
The solution
You have entered
[src]
oo ____ \ ` \ /1 \ \ sin|--| / | n| / \3 / /___, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{1}{3^{n}} \right)}$$
Sum(sin(1/(3^n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin{\left(\frac{1}{3^{n}} \right)}$$
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
$$a_{n} = \sin{\left(3^{- n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(3^{- n} \right)}}{\sin{\left(3^{- (n + 1)} \right)}}}\right|$$
Let's take the limit
we find
$$\sin{\left(\frac{1}{3^{n}} \right)}$$
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
$$a_{n} = \sin{\left(3^{- n} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(3^{- n} \right)}}{\sin{\left(3^{- (n + 1)} \right)}}}\right|$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ / -n\ / sin\3 / /__, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(3^{- n} \right)}$$
Sum(sin(3^(-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