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((4n-1)/(7n+4))^n^2

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  • Sum of series:
  • (3^n-1)/n! (3^n-1)/n!
  • sin2n sin2n
  • (5^n-4^n)/6^n (5^n-4^n)/6^n
  • 2i 2i

  • Identical expressions

  • ((four n- one)/(7n+4))^n^ two
  • ((4n minus 1) divide by (7n plus 4)) to the power of n squared
  • ((four n minus one) divide by (7n plus 4)) to the power of n to the power of two
  • ((4n-1)/(7n+4))n2
  • 4n-1/7n+4n2
  • ((4n-1)/(7n+4))^n²
  • ((4n-1)/(7n+4)) to the power of n to the power of 2
  • 4n-1/7n+4^n^2
  • ((4n-1) divide by (7n+4))^n^2

  • Similar expressions

  • ((4n+1)/(7n+4))^n^2
  • ((4n-1)/(7n-4))^n^2
Sum of series/ ((4n-1)/(7n+4))^n^2

Sum of series ((4n-1)/(7n+4))^n^2


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The solution

You have entered [src] 
  oo               
____               
\   `              
 \             / 2\
  \            \n /
   )  /4*n - 1\    
  /   |-------|    
 /    \7*n + 4/    
/___,              
n = 1
$$\sum_{n=1}^{\infty} \left(\frac{4 n - 1}{7 n + 4}\right)^{n^{2}}$$ 
Sum(((4*n - 1)/(7*n + 4))^(n^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{4 n - 1}{7 n + 4}\right)^{n^{2}}$$
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
$$a_{n} = \left(\frac{4 n - 1}{7 n + 4}\right)^{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{4 n + 3}{7 n + 11}\right)^{- \left(n + 1\right)^{2}} \left|{\left(\frac{4 n - 1}{7 n + 4}\right)^{n^{2}}}\right|\right)$$
Let's take the limit
we find
False

False
The rate of convergence of the power series
Numerical answer [src] 
0.296222737937373026295040536044
0.296222737937373026295040536044
The graph
Sum of series ((4n-1)/(7n+4))^n^2

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