Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
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(3*n/(9*n-1))^n
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factorial(n)/((n*n))
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factorial(n)/(n+1)
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1/factorial(n^1)
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Identical expressions
- (three *n/(nine *n- one))^n
- (3 multiply by n divide by (9 multiply by n minus 1)) to the power of n
- (three multiply by n divide by (nine multiply by n minus one)) to the power of n
- (3*n/(9*n-1))n
- 3*n/9*n-1n
- (3n/(9n-1))^n
- (3n/(9n-1))n
- 3n/9n-1n
- 3n/9n-1^n
- (3*n divide by (9*n-1))^n
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Similar expressions
- (3*n/(9*n+1))^n
Sum of series (3*n/(9*n-1))^n
The solution
You have entered
[src]
oo ____ \ ` \ n \ / 3*n \ / |-------| / \9*n - 1/ /___, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{3 n}{9 n - 1}\right)^{n}$$
Sum(((3*n)/(9*n - 1))^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{3 n}{9 n - 1}\right)^{n}$$
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
$$a_{n} = \left(\frac{3 n}{9 n - 1}\right)^{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{3 \left(n + 1\right)}{9 n + 8}\right)^{- n - 1} \left|{\left(\frac{3 n}{9 n - 1}\right)^{n}}\right|\right)$$
Let's take the limit
we find
$$\left(\frac{3 n}{9 n - 1}\right)^{n}$$
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
$$a_{n} = \left(\frac{3 n}{9 n - 1}\right)^{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{3 \left(n + 1\right)}{9 n + 8}\right)^{- n - 1} \left|{\left(\frac{3 n}{9 n - 1}\right)^{n}}\right|\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ n \ / 3*n \ / |--------| / \-1 + 9*n/ /___, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{3 n}{9 n - 1}\right)^{n}$$
Sum((3*n/(-1 + 9*n))^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