Mister Exam
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How to use it?
- Sum of series:
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sin1/n
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28
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0.5^n
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((n+1)^n*3^(2n-1))/(2n+1)!
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Identical expressions
- pi/ three ^(2n+ one)/(2n+ one)!
- Pi divide by 3 to the power of (2n plus 1) divide by (2n plus 1)!
- Pi divide by three to the power of (2n plus one) divide by (2n plus one)!
- pi/3(2n+1)/(2n+1)!
- pi/32n+1/2n+1!
- pi/3^2n+1/2n+1!
- pi divide by 3^(2n+1) divide by (2n+1)!
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Similar expressions
- pi/3^(2n-1)/(2n+1)!
- pi/3^(2n+1)/(2n-1)!
Sum of series pi/3^(2n+1)/(2n+1)!
The solution
You have entered
[src]
oo _____ \ ` \ / pi \ \ |--------| \ | 2*n + 1| / \3 / / ---------- / (2*n + 1)! /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\pi \frac{1}{3^{2 n + 1}}}{\left(2 n + 1\right)!}$$
Sum((pi/3^(2*n + 1))/factorial(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\pi \frac{1}{3^{2 n + 1}}}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \frac{3^{- 2 n - 1} \pi}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- 2 n - 1} \cdot 3^{2 n + 3} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|\right)$$
Let's take the limit
we find
$$\frac{\pi \frac{1}{3^{2 n + 1}}}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \frac{3^{- 2 n - 1} \pi}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- 2 n - 1} \cdot 3^{2 n + 3} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
pi*(-1 + 3*sinh(1/3))
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3
$$\frac{\pi \left(-1 + 3 \sinh{\left(\frac{1}{3} \right)}\right)}{3}$$
pi*(-1 + 3*sinh(1/3))/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