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2^(2*n-1)/9^n

  • How to use it?

  • Sum of series:
  • (n-1)/2^(n+1) (n-1)/2^(n+1)
  • 6/(n^2-10n+24) 6/(n^2-10n+24)
  • a^n
  • 2^(2*n-1)/9^n 2^(2*n-1)/9^n

  • Identical expressions

  • two ^(two *n- one)/ nine ^n
  • 2 to the power of (2 multiply by n minus 1) divide by 9 to the power of n
  • two to the power of (two multiply by n minus one) divide by nine to the power of n
  • 2(2*n-1)/9n
  • 22*n-1/9n
  • 2^(2n-1)/9^n
  • 2(2n-1)/9n
  • 22n-1/9n
  • 2^2n-1/9^n
  • 2^(2*n-1) divide by 9^n

  • Similar expressions

  • 2^(2*n+1)/9^n
Sum of series/ 2^(2*n-1)/9^n

Sum of series 2^(2*n-1)/9^n


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The solution

You have entered [src] 
  oo          
____          
\   `         
 \     2*n - 1
  \   2       
   )  --------
  /       n   
 /       9    
/___,         
n = 0
$$\sum_{n=0}^{\infty} \frac{2^{2 n - 1}}{9^{n}}$$ 
Sum(2^(2*n - 1)/9^n, (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{2^{2 n - 1}}{9^{n}}$$
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
$$a_{n} = 2^{2 n - 1}$$
and
$$x_{0} = -9$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-9 + \lim_{n \to \infty}\left(2^{- 2 n - 1} \cdot 2^{2 n - 1}\right)\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
9/10
$$\frac{9}{10}$$ 
9/10
Numerical answer [src] 
0.900000000000000000000000000000
0.900000000000000000000000000000
The graph
Sum of series 2^(2*n-1)/9^n

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