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*sqrt(n)/(n+100)
- n^2*x^n
-
2(1/(n^2+5n+6))
- ((-1)^n)*((x^n)/(2n*n!))
-
Identical expressions
- one /((5n- four)(5n+ one))
- 1 divide by ((5n minus 4)(5n plus 1))
- one divide by ((5n minus four)(5n plus one))
- 1/5n-45n+1
- 1 divide by ((5n-4)(5n+1))
-
Similar expressions
- (1)/5*n-4*5*n+1
- 1/((5n+4)(5n+1))
- 1/((5n-4)(5n-1))
Sum of series 1/((5n-4)(5n+1))
The solution
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[src]
oo ___ \ ` \ 1 ) ------------------- / (5*n - 4)*(5*n + 1) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
Sum(1/((5*n - 4)*(5*n + 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
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
$$a_{n} = \frac{1}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 6\right) \left|{\frac{1}{5 n - 4}}\right|\right)$$
Let's take the limit
we find
$$\frac{1}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
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
$$a_{n} = \frac{1}{\left(5 n - 4\right) \left(5 n + 1\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5 n + 6\right) \left|{\frac{1}{5 n - 4}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
Gamma(11/5) ------------ 6*Gamma(6/5)
$$\frac{\Gamma\left(\frac{11}{5}\right)}{6 \Gamma\left(\frac{6}{5}\right)}$$
gamma(11/5)/(6*gamma(6/5))
The graph
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