Mister Exam

Other calculators

  • How to use it?

  • Sum of series:
  • log(2*x-4)/(n^2-x^2)
  • ln(n/(n+1)) ln(n/(n+1))
  • 1/(ln(x+2)*ln(x+2))
  • sin(kx/n)*x/n

  • Identical expressions

  • log(two *x- four)/(n^ two -x^ two)
  • logarithm of (2 multiply by x minus 4) divide by (n squared minus x squared )
  • logarithm of (two multiply by x minus four) divide by (n to the power of two minus x to the power of two)
  • log(2*x-4)/(n2-x2)
  • log2*x-4/n2-x2
  • log(2*x-4)/(n²-x²)
  • log(2*x-4)/(n to the power of 2-x to the power of 2)
  • log(2x-4)/(n^2-x^2)
  • log(2x-4)/(n2-x2)
  • log2x-4/n2-x2
  • log2x-4/n^2-x^2
  • log(2*x-4) divide by (n^2-x^2)

  • Similar expressions

  • log(2*x+4)/(n^2-x^2)
  • log(2*x-4)/(n^2+x^2)
Sum of series/ log(2*x-4)/(n^2-x^2)

Sum of series log(2*x-4)/(n^2-x^2)


∑
=

The solution

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

False
The answer [src] 
 //     2\                         /      2\                      \               
 |\1 - x /*lerchphi(1, 1, 1 - x)   \-1 + x /*lerchphi(1, 1, 1 + x)|               
-|------------------------------ + -------------------------------|*log(-4 + 2*x) 
 \             2*x                               2*x              /               
----------------------------------------------------------------------------------
                                 (1 + x)*(-1 + x)
$$- \frac{\left(\frac{\left(1 - x^{2}\right) \Phi\left(1, 1, 1 - x\right)}{2 x} + \frac{\left(x^{2} - 1\right) \Phi\left(1, 1, x + 1\right)}{2 x}\right) \log{\left(2 x - 4 \right)}}{\left(x - 1\right) \left(x + 1\right)}$$ 
-((1 - x^2)*lerchphi(1, 1, 1 - x)/(2*x) + (-1 + x^2)*lerchphi(1, 1, 1 + x)/(2*x))*log(-4 + 2*x)/((1 + x)*(-1 + x))

    Examples of finding the sum of a series

    © Mister Exam – Calculator