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  • sin^ two (x)/ three ^n
  • sinus of squared (x) divide by 3 to the power of n
  • sinus of to the power of two (x) divide by three to the power of n
  • sin2(x)/3n
  • sin2x/3n
  • sin²(x)/3^n
  • sin to the power of 2(x)/3 to the power of n
  • sin^2x/3^n
  • sin^2(x) divide by 3^n
Sum of series/ sin^2(x)/3^n

Sum of series sin^2(x)/3^n


∑
=

The solution

You have entered [src] 
  oo         
____         
\   `        
 \       2   
  \   sin (x)
   )  -------
  /       n  
 /       3   
/___,        
n = 1
∑n=1∞sin⁡2(x)3n\sum_{n=1}^{\infty} \frac{\sin^{2}{\left(x \right)}}{3^{n}} 
Sum(sin(x)^2/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
sin⁡2(x)3n\frac{\sin^{2}{\left(x \right)}}{3^{n}}
It is a series of species
an(cx−x0)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+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=sin⁡2(x)a_{n} = \sin^{2}{\left(x \right)}
and
x0=−3x_{0} = -3
,
d=−1d = -1
,
c=0c = 0
then
1R=∞~(−3+lim⁡n→∞1)\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{n \to \infty} 1\right)
Let's take the limit
we find
False

R=0R = 0
The answer [src] 
   2   
sin (x)
-------
   2
sin⁡2(x)2\frac{\sin^{2}{\left(x \right)}}{2} 
sin(x)^2/2

    Examples of finding the sum of a series

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