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  • Sum of series:
  • sin(x/2^n)*cos(3*x/2^n)
  • factorial(n)*factorial(2*n+1)/factorial(3*n) factorial(n)*factorial(2*n+1)/factorial(3*n)
  • factorial(2*n+1)/factorial(3*n-1) factorial(2*n+1)/factorial(3*n-1)
  • (9*n-7)/2^n (9*n-7)/2^n

  • Identical expressions

  • tg(six /(two n)^(one / three))^2*(x- twenty-four)^n
  • tg(6 divide by (2n) to the power of (1 divide by 3)) squared multiply by (x minus 24) to the power of n
  • tg(six divide by (two n) to the power of (one divide by three)) squared multiply by (x minus twenty minus four) to the power of n
  • tg(6/(2n)(1/3))2*(x-24)n
  • tg6/2n1/32*x-24n
  • tg(6/(2n)^(1/3))²*(x-24)^n
  • tg(6/(2n) to the power of (1/3)) to the power of 2*(x-24) to the power of n
  • tg(6/(2n)^(1/3))^2(x-24)^n
  • tg(6/(2n)(1/3))2(x-24)n
  • tg6/2n1/32x-24n
  • tg6/2n^1/3^2x-24^n
  • tg(6 divide by (2n)^(1 divide by 3))^2*(x-24)^n

  • Similar expressions

  • tg(6/(2n)^(1/3))^2*(x+24)^n
Sum of series/ tg(6/(2n)^(1/3))^2*(x-24)^n

Sum of series tg(6/(2n)^(1/3))^2*(x-24)^n


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The solution

You have entered [src] 
  oo                         
____                         
\   `                        
 \       2/   6   \         n
  \   tan |-------|*(x - 24) 
  /       |3 _____|          
 /        \\/ 2*n /          
/___,                        
n = 1
$$\sum_{n=1}^{\infty} \left(x - 24\right)^{n} \tan^{2}{\left(\frac{6}{\sqrt[3]{2 n}} \right)}$$ 
Sum(tan(6/(2*n)^(1/3))^2*(x - 24)^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(x - 24\right)^{n} \tan^{2}{\left(\frac{6}{\sqrt[3]{2 n}} \right)}$$
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
$$a_{n} = \tan^{2}{\left(\frac{3 \cdot 2^{\frac{2}{3}}}{\sqrt[3]{n}} \right)}$$
and
$$x_{0} = 24$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = 24 + \lim_{n \to \infty}\left(\tan^{2}{\left(\frac{3 \cdot 2^{\frac{2}{3}}}{\sqrt[3]{n}} \right)} \left|{\frac{1}{\tan^{2}{\left(\frac{3 \cdot 2^{\frac{2}{3}}}{\sqrt[3]{n + 1}} \right)}}}\right|\right)$$
Let's take the limit
we find
$$R = 25$$
The answer [src] 
  oo                         
____                         
\   `                        
 \                   /   2/3\
  \            n    2|3*2   |
   )  (-24 + x) *tan |------|
  /                  |3 ___ |
 /                   \\/ n  /
/___,                        
n = 1
$$\sum_{n=1}^{\infty} \left(x - 24\right)^{n} \tan^{2}{\left(\frac{3 \cdot 2^{\frac{2}{3}}}{\sqrt[3]{n}} \right)}$$ 
Sum((-24 + x)^n*tan(3*2^(2/3)/n^(1/3))^2, (n, 1, oo))

    Examples of finding the sum of a series

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