Mister Exam

Other calculators

  • How to use it?

  • Sum of series:
  • 2i+1 2i+1
  • 5000n+60000 5000n+60000
  • 1/(n^2-3n+2) 1/(n^2-3n+2)
  • sin(n*x)/(n^2+1)

  • Identical expressions

  • sin(n*x)/(n^ two + one)
  • sinus of (n multiply by x) divide by (n squared plus 1)
  • sinus of (n multiply by x) divide by (n to the power of two plus one)
  • sin(n*x)/(n2+1)
  • sinn*x/n2+1
  • sin(n*x)/(n²+1)
  • sin(n*x)/(n to the power of 2+1)
  • sin(nx)/(n^2+1)
  • sin(nx)/(n2+1)
  • sinnx/n2+1
  • sinnx/n^2+1
  • sin(n*x) divide by (n^2+1)

  • Similar expressions

  • sin(n*x)/(n^2-1)
Sum of series/ sin(n*x)/(n^2+1)

Sum of series sin(n*x)/(n^2+1)


∑
=

The solution

You have entered [src] 
  oo          
____          
\   `         
 \    sin(n*x)
  \   --------
  /     2     
 /     n  + 1 
/___,         
n = 1
∑n=1∞sin⁡(nx)n2+1\sum_{n=1}^{\infty} \frac{\sin{\left(n x \right)}}{n^{2} + 1} 
Sum(sin(n*x)/(n^2 + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
sin⁡(nx)n2+1\frac{\sin{\left(n x \right)}}{n^{2} + 1}
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⁡(nx)n2+1a_{n} = \frac{\sin{\left(n x \right)}}{n^{2} + 1}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(((n+1)2+1)∣sin⁡(nx)sin⁡(x(n+1))∣n2+1)1 = \lim_{n \to \infty}\left(\frac{\left(\left(n + 1\right)^{2} + 1\right) \left|{\frac{\sin{\left(n x \right)}}{\sin{\left(x \left(n + 1\right) \right)}}}\right|}{n^{2} + 1}\right)
Let's take the limit
we find
True

False

    Examples of finding the sum of a series

    © Mister Exam – Calculator