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
$$R = 1$$
$$\frac{f x^{n}}{3 \left(n + 4\right)}$$
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))