Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
-
(n+1)^2/2^(n-1)
-
(-1)^n/n
-
(5/6)^n
-
28
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Identical expressions
- one /(3n+ two)(3n- one)
- 1 divide by (3n plus 2)(3n minus 1)
- one divide by (3n plus two)(3n minus one)
- 1/3n+23n-1
- 1 divide by (3n+2)(3n-1)
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Similar expressions
- 1/(3n+2)(3n+1)
- 1/(3n-2)(3n-1)
Sum of series 1/(3n+2)(3n-1)
The solution
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oo ___ \ ` \ 3*n - 1 ) ------- / 3*n + 2 /__, n = 1
$$\sum_{n=1}^{\infty} \frac{3 n - 1}{3 n + 2}$$
Sum((3*n - 1)/(3*n + 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 n - 1}{3 n + 2}$$
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
$$a_{n} = \frac{3 n - 1}{3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 5\right) \left|{3 n - 1}\right|}{\left(3 n + 2\right)^{2}}\right)$$
Let's take the limit
we find
$$\frac{3 n - 1}{3 n + 2}$$
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
$$a_{n} = \frac{3 n - 1}{3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 5\right) \left|{3 n - 1}\right|}{\left(3 n + 2\right)^{2}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
Numerical answer
The series diverges
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