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  • Sum of series:
  • (2/9)^n
  • n^2/n!
  • (2/3)^n
  • (-1)^(n+1)/n
  • Identical expressions

  • f*x^n/(three (n+ four))
  • f multiply by x to the power of n divide by (3(n plus 4))
  • f multiply by x to the power of n divide by (three (n plus four))
  • f*xn/(3(n+4))
  • f*xn/3n+4
  • fx^n/(3(n+4))
  • fxn/(3(n+4))
  • fxn/3n+4
  • fx^n/3n+4
  • f*x^n divide by (3(n+4))
  • Similar expressions

  • f*x^n/(3(n-4))

Sum of series f*x^n/(3(n+4))



=

The solution

You have entered [src]
  oo           
____           
\   `          
 \          n  
  \      f*x   
  /   ---------
 /    3*(n + 4)
/___,          
n = 1          
$$\sum_{n=1}^{\infty} \frac{f x^{n}}{3 \left(n + 4\right)}$$
Sum((f*x^n)/((3*(n + 4))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{f x^{n}}{3 \left(n + 4\right)}$$
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
$$a_{n} = \frac{f}{3 n + 12}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{3 n + 15}{3 n + 12}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer [src]
  //  /              2      3               \                         \
  ||  |     5*x   5*x    5*x                |                         |
  ||  |-5 - --- - ---- - ----               |                         |
  ||  |      2     3      4     5*log(1 - x)|                         |
  ||x*|---------------------- - ------------|                         |
  ||  |           4                   5     |                         |
  ||  \          x                   x      /                         |
  ||-----------------------------------------  for And(x >= -1, x < 1)|
  ||                    15                                            |
f*|<                                                                  |
  ||               oo                                                 |
  ||             ____                                                 |
  ||             \   `                                                |
  ||              \        n                                          |
  ||               \      x                                           |
  ||               /   --------                       otherwise       |
  ||              /    12 + 3*n                                       |
  ||             /___,                                                |
  \\             n = 1                                                /
$$f \left(\begin{cases} \frac{x \left(\frac{- \frac{5 x^{3}}{4} - \frac{5 x^{2}}{3} - \frac{5 x}{2} - 5}{x^{4}} - \frac{5 \log{\left(1 - x \right)}}{x^{5}}\right)}{15} & \text{for}\: x \geq -1 \wedge x < 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{3 n + 12} & \text{otherwise} \end{cases}\right)$$
f*Piecewise((x*((-5 - 5*x/2 - 5*x^2/3 - 5*x^3/4)/x^4 - 5*log(1 - x)/x^5)/15, (x >= -1)∧(x < 1)), (Sum(x^n/(12 + 3*n), (n, 1, oo)), True))
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