Mister Exam
Other calculators
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- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
n^2
-
-1^(n-1)/((2*n-1)*3^(n-1))
-
(5/2^n+1)+ln(n/n+1)
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1/((3*n-1)*(3*n+2))
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Identical expressions
- - one ^(n- one)/((two *n- one)* three ^(n- one))
- minus 1 to the power of (n minus 1) divide by ((2 multiply by n minus 1) multiply by 3 to the power of (n minus 1))
- minus one to the power of (n minus one) divide by ((two multiply by n minus one) multiply by three to the power of (n minus one))
- -1(n-1)/((2*n-1)*3(n-1))
- -1n-1/2*n-1*3n-1
- -1^(n-1)/((2n-1)3^(n-1))
- -1(n-1)/((2n-1)3(n-1))
- -1n-1/2n-13n-1
- -1^n-1/2n-13^n-1
- -1^(n-1) divide by ((2*n-1)*3^(n-1))
-
Similar expressions
- -1^(n-1)/((2*n+1)*3^(n-1))
- -1^(n+1)/((2*n-1)*3^(n-1))
- 1^(n-1)/((2*n-1)*3^(n-1))
- -1^(n-1)/((2*n-1)*3^(n+1))
Sum of series -1^(n-1)/((2*n-1)*3^(n-1))
The solution
You have entered
[src]
oo ____ \ ` \ n - 1 \ -1 ) ---------------- / n - 1 / (2*n - 1)*3 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right) 1^{n - 1}}{3^{n - 1} \left(2 n - 1\right)}$$
Sum((-1^(n - 1))/(((2*n - 1)*3^(n - 1))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right) 1^{n - 1}}{3^{n - 1} \left(2 n - 1\right)}$$
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
$$a_{n} = - \frac{3^{1 - n}}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{n} 3^{1 - n} \left(2 n + 1\right) \left|{\frac{1}{2 n - 1}}\right|\right)$$
Let's take the limit
we find
$$\frac{\left(-1\right) 1^{n - 1}}{3^{n - 1} \left(2 n - 1\right)}$$
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
$$a_{n} = - \frac{3^{1 - n}}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{n} 3^{1 - n} \left(2 n + 1\right) \left|{\frac{1}{2 n - 1}}\right|\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
/ ___\
___ |\/ 3 |
-\/ 3 *atanh|-----|
\ 3 /
$$- \sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3}}{3} \right)}$$
-sqrt(3)*atanh(sqrt(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