Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
((-1)^(n+1))/(2^n)
- (-1)^n*x^n/(2*n+1)
- (x^n)/(n*(n+1))
- k*x+b
-
Identical expressions
- (x^n)/(n*(n+ one))
- (x to the power of n) divide by (n multiply by (n plus 1))
- (x to the power of n) divide by (n multiply by (n plus one))
- (xn)/(n*(n+1))
- xn/n*n+1
- (x^n)/(n(n+1))
- (xn)/(n(n+1))
- xn/nn+1
- x^n/nn+1
- (x^n) divide by (n*(n+1))
-
Similar expressions
- (x^n)/(n*(n-1))
Sum of series (x^n)/(n*(n+1))
The solution
You have entered
[src]
oo ____ \ ` \ n \ x / --------- / n*(n + 1) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{x^{n}}{n \left(n + 1\right)}$$
Sum(x^n/((n*(n + 1))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x^{n}}{n \left(n + 1\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n + 2}{n}\right)$$
Let's take the limit
we find
$$R = 1$$
$$\frac{x^{n}}{n \left(n + 1\right)}$$
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
$$a_{n} = \frac{1}{n \left(n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{n + 2}{n}\right)$$
Let's take the limit
we find
$$R = 1$$
The answer
[src]
/ /2 (2 - 2*x)*log(1 - x)\ |x*|- + --------------------| | |x 2 | | \ x / |---------------------------- for |x| <= 1 | 2 | | oo < ____ | \ ` | \ n | \ x | ) ------ otherwise | / 2 | / n + n | /___, \ n = 1
$$\begin{cases} \frac{x \left(\frac{2}{x} + \frac{\left(2 - 2 x\right) \log{\left(1 - x \right)}}{x^{2}}\right)}{2} & \text{for}\: \left|{x}\right| \leq 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2} + n} & \text{otherwise} \end{cases}$$
Piecewise((x*(2/x + (2 - 2*x)*log(1 - x)/x^2)/2, |x| <= 1), (Sum(x^n/(n + n^2), (n, 1, oo)), True))
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n