Mister Exam

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  • Sum of series:
  • n n
  • sin(x/n^2)
  • (3n+4)/(n/((n+1)(n+2))) (3n+4)/(n/((n+1)(n+2)))
  • sin(3n) sin(3n)
  • Identical expressions

  • sin(x/n^ two)
  • sinus of (x divide by n squared )
  • sinus of (x divide by n to the power of two)
  • sin(x/n2)
  • sinx/n2
  • sin(x/n²)
  • sin(x/n to the power of 2)
  • sinx/n^2
  • sin(x divide by n^2)

Sum of series sin(x/n^2)



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The solution

You have entered [src]
  oo         
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\   `        
 \       /x \
  \   sin|--|
  /      | 2|
 /       \n /
/___,        
n = 1        
n=1sin(xn2)\sum_{n=1}^{\infty} \sin{\left(\frac{x}{n^{2}} \right)}
Sum(sin(x/n^2), (n, 1, oo))
The radius of convergence of the power series
Given number:
sin(xn2)\sin{\left(\frac{x}{n^{2}} \right)}
It is a series of species
an(cxx0)dna_{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:
Rd=x0+limnanan+1cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=sin(xn2)a_{n} = \sin{\left(\frac{x}{n^{2}} \right)}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=limnsin(xn2)sin(x(n+1)2)1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{x}{n^{2}} \right)}}{\sin{\left(\frac{x}{\left(n + 1\right)^{2}} \right)}}}\right|
Let's take the limit
we find
True

False

    Examples of finding the sum of a series