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  • Sum of series:
  • (2^n+(-1)^n)/5^n (2^n+(-1)^n)/5^n
  • (-1)^n*sqrt(n)/(n+100) (-1)^n*sqrt(n)/(n+100)
  • x^n/(1-x^n)
  • 762.5*(0.525*0.475)^n 762.5*(0.525*0.475)^n

  • Identical expressions

  • three *(five *x- three)^n/((five ^n* four))
  • 3 multiply by (5 multiply by x minus 3) to the power of n divide by ((5 to the power of n multiply by 4))
  • three multiply by (five multiply by x minus three) to the power of n divide by ((five to the power of n multiply by four))
  • 3*(5*x-3)n/((5n*4))
  • 3*5*x-3n/5n*4
  • 3(5x-3)^n/((5^n4))
  • 3(5x-3)n/((5n4))
  • 35x-3n/5n4
  • 35x-3^n/5^n4
  • 3*(5*x-3)^n divide by ((5^n*4))

  • Similar expressions

  • 3*(5*x+3)^n/((5^n*4))
Sum of series/ 3*(5*x-3)^n/((5^n*4))

Sum of series 3*(5*x-3)^n/((5^n*4))


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The solution

You have entered [src] 
  oo              
____              
\   `             
 \               n
  \   3*(5*x - 3) 
   )  ------------
  /        n      
 /        5 *4    
/___,             
n = 1
$$\sum_{n=1}^{\infty} \frac{3 \left(5 x - 3\right)^{n}}{4 \cdot 5^{n}}$$ 
Sum((3*(5*x - 3)^n)/((5^n*4)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 \left(5 x - 3\right)^{n}}{4 \cdot 5^{n}}$$
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
$$a_{n} = \frac{3 \cdot 5^{- n}}{4}$$
and
$$x_{0} = 3$$
,
$$d = 1$$
,
$$c = 5$$
then
$$R = \frac{3 + \lim_{n \to \infty}\left(5^{- n} 5^{n + 1}\right)}{5}$$
Let's take the limit
we find
$$R = \frac{8}{5}$$
The answer [src] 
  //      -3/5 + x                           \
  ||      --------         for |-3/5 + x| < 1|
  ||      8/5 - x                            |
  ||                                         |
  ||  oo                                     |
3*|< ___                                     |
  || \  `                                    |
  ||  \    -n           n                    |
  ||  /   5  *(-3 + 5*x)       otherwise     |
  || /__,                                    |
  \\n = 1                                    /
----------------------------------------------
                      4
$$\frac{3 \left(\begin{cases} \frac{x - \frac{3}{5}}{\frac{8}{5} - x} & \text{for}\: \left|{x - \frac{3}{5}}\right| < 1 \\\sum_{n=1}^{\infty} 5^{- n} \left(5 x - 3\right)^{n} & \text{otherwise} \end{cases}\right)}{4}$$ 
3*Piecewise(((-3/5 + x)/(8/5 - x), |-3/5 + x| < 1), (Sum(5^(-n)*(-3 + 5*x)^n, (n, 1, oo)), True))/4

    Examples of finding the sum of a series

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