Mister Exam
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- 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:
-
sin((n^2+3)/(n^3+2))^2
-
factorial(n+1)/((7^n*n))
-
n^(n+1)/factorial(n+1)
- ln(1+x/2)
-
Identical expressions
- three /((4n- one)*(4n+ three))
- 3 divide by ((4n minus 1) multiply by (4n plus 3))
- three divide by ((4n minus one) multiply by (4n plus three))
- 3/((4n-1)(4n+3))
- 3/4n-14n+3
- 3 divide by ((4n-1)*(4n+3))
-
Similar expressions
- 3/(4n-1)(4n+3)
- 3/((4n+1)*(4n+3))
- 3/((4n-1)(4n+3))
- 3/((4n-1)*(4n-3))
Sum of series 3/((4n-1)*(4n+3))
The solution
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[src]
oo ___ \ ` \ 3 ) ------------------- / (4*n - 1)*(4*n + 3) /__, n = 1
$$\sum_{n=1}^{\infty} \frac{3}{\left(4 n - 1\right) \left(4 n + 3\right)}$$
Sum(3/(((4*n - 1)*(4*n + 3))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3}{\left(4 n - 1\right) \left(4 n + 3\right)}$$
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
$$a_{n} = \frac{3}{\left(4 n - 1\right) \left(4 n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(4 n + 7\right) \left|{\frac{1}{4 n - 1}}\right|\right)$$
Let's take the limit
we find
$$\frac{3}{\left(4 n - 1\right) \left(4 n + 3\right)}$$
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
$$a_{n} = \frac{3}{\left(4 n - 1\right) \left(4 n + 3\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(4 n + 7\right) \left|{\frac{1}{4 n - 1}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
Gamma(11/4) ------------ 7*Gamma(7/4)
$$\frac{\Gamma\left(\frac{11}{4}\right)}{7 \Gamma\left(\frac{7}{4}\right)}$$
gamma(11/4)/(7*gamma(7/4))
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