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  • Sum of series:
  • nx^n/(8n-9)*3^n
  • sqrt(n+1)-sqrt(n)/sqrt(n^2+n) sqrt(n+1)-sqrt(n)/sqrt(n^2+n)
  • (3*x^2-2*x+3)/factorial(x+1) (3*x^2-2*x+3)/factorial(x+1)
  • sin(3*n)/(n+2) sin(3*n)/(n+2)

  • Identical expressions

  • x^ two *arctg(x^ two / four)
  • x squared multiply by arctg(x squared divide by 4)
  • x to the power of two multiply by arctg(x to the power of two divide by four)
  • x2*arctg(x2/4)
  • x2*arctgx2/4
  • x²*arctg(x²/4)
  • x to the power of 2*arctg(x to the power of 2/4)
  • x^2arctg(x^2/4)
  • x2arctg(x2/4)
  • x2arctgx2/4
  • x^2arctgx^2/4
  • x^2*arctg(x^2 divide by 4)
Sum of series/ x^2*arctg(x^2/4)

Sum of series x^2*arctg(x^2/4)


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The solution

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

False
The answer [src] 
          / 2\
    2     |x |
oo*x *atan|--|
          \4 /
$$\infty x^{2} \operatorname{atan}{\left(\frac{x^{2}}{4} \right)}$$ 
oo*x^2*atan(x^2/4)

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