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-cos(pi/n)
- ((-1)^n*(3*cos(n*x)-pi*n*sin(n*x))-3*cos(n*x+pi*n/3))/n^2
- sin(x/2^n)*cos(3*x/2^n)
- sin(n*x)/(n^2+1)
-
Identical expressions
- sin(n*x)/(n^ two + one)
- sinus of (n multiply by x) divide by (n squared plus 1)
- sinus of (n multiply by x) divide by (n to the power of two plus one)
- sin(n*x)/(n2+1)
- sinn*x/n2+1
- sin(n*x)/(n²+1)
- sin(n*x)/(n to the power of 2+1)
- sin(nx)/(n^2+1)
- sin(nx)/(n2+1)
- sinnx/n2+1
- sinnx/n^2+1
- sin(n*x) divide by (n^2+1)
-
Similar expressions
- sin(n*x)/(n^2-1)
Sum of series sin(n*x)/(n^2+1)
The solution
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oo ____ \ ` \ sin(n*x) \ -------- / 2 / n + 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(n x \right)}}{n^{2} + 1}$$
Sum(sin(n*x)/(n^2 + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(n x \right)}}{n^{2} + 1}$$
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
$$a_{n} = \frac{\sin{\left(n x \right)}}{n^{2} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \left|{\frac{\sin{\left(n x \right)}}{\sin{\left(x \left(n + 1\right) \right)}}}\right|}{n^{2} + 1}\right)$$
Let's take the limit
we find
$$\frac{\sin{\left(n x \right)}}{n^{2} + 1}$$
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
$$a_{n} = \frac{\sin{\left(n x \right)}}{n^{2} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \left|{\frac{\sin{\left(n x \right)}}{\sin{\left(x \left(n + 1\right) \right)}}}\right|}{n^{2} + 1}\right)$$
Let's take the limit
we find
True
False
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