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(3*n/(9*n-1))^n

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  • Sum of series:
  • (3*n/(9*n-1))^n (3*n/(9*n-1))^n
  • factorial(n)/((n*n)) factorial(n)/((n*n))
  • arctg(1/(n^2+n+1)) arctg(1/(n^2+n+1))
  • (ln(n)/n)^p

  • 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

  • Similar expressions

  • (3*n/(9*n+1))^n
Sum of series/ (3*n/(9*n-1))^n

Sum of series (3*n/(9*n-1))^n


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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
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))
Numerical answer [src] 
0.561769005363273344436983043632
0.561769005363273344436983043632
The graph
Sum of series (3*n/(9*n-1))^n

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