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:
- (6x-1)^n/((n+1)*6^n)
- chx/(shx^2+1)
-
(2/3)^n
-
1/ln(n+1)
-
Identical expressions
- chx/(shx^ two + one)
- chx divide by (shx squared plus 1)
- chx divide by (shx to the power of two plus one)
- chx/(shx2+1)
- chx/shx2+1
- chx/(shx²+1)
- chx/(shx to the power of 2+1)
- chx/shx^2+1
- chx divide by (shx^2+1)
-
Similar expressions
- chx/(shx^2-1)
Sum of series chx/(shx^2+1)
The solution
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[src]
oo ____ \ ` \ cosh(x) \ ------------ / 2 / sinh (x) + 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
Sum(cosh(x)/(sinh(x)^2 + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
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
$$a_{n} = \frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$\frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
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
$$a_{n} = \frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
True
False
The answer
[src]
oo*cosh(x)
------------
2
1 + sinh (x)
$$\frac{\infty \cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)} + 1}$$
oo*cosh(x)/(1 + sinh(x)^2)
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