Mister Exam

### Other calculators

• #### How to use it?

• 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
$$R = 1$$
  //  /              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))
To see a detailed solution - share to all your student friends
To see a detailed solution,
share to all your student friends: