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:
-
(3^n-1)/n!
-
sin2n
-
(5^n-4^n)/6^n
-
2i
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Identical expressions
- one /((3n+ one)(3n+ two))
- 1 divide by ((3n plus 1)(3n plus 2))
- one divide by ((3n plus one)(3n plus two))
- 1/3n+13n+2
- 1 divide by ((3n+1)(3n+2))
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Similar expressions
- 1/((3n-1)(3n+2))
- 1/(3n+1)(3n+2)
- 1/((3n+1)(3n-2))
Sum of series 1/((3n+1)(3n+2))
The solution
You have entered
[src]
oo ___ \ ` \ 1 ) ------------------- / (3*n + 1)*(3*n + 2) /__, n = 0
$$\sum_{n=0}^{\infty} \frac{1}{\left(3 n + 1\right) \left(3 n + 2\right)}$$
Sum(1/((3*n + 1)*(3*n + 2)), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(3 n + 1\right) \left(3 n + 2\right)}$$
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
$$a_{n} = \frac{1}{\left(3 n + 1\right) \left(3 n + 2\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 4\right) \left(3 n + 5\right)}{\left(3 n + 1\right) \left(3 n + 2\right)}\right)$$
Let's take the limit
we find
$$\frac{1}{\left(3 n + 1\right) \left(3 n + 2\right)}$$
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
$$a_{n} = \frac{1}{\left(3 n + 1\right) \left(3 n + 2\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 4\right) \left(3 n + 5\right)}{\left(3 n + 1\right) \left(3 n + 2\right)}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
-2*pi*I / 2*pi*I\ 2*pi*I / 4*pi*I\ -2*pi*I / 4*pi*I\ 2*pi*I / 2*pi*I\
------- | ------| ------ | ------| ------- | ------| ------ | ------|
3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
e *log\1 - e / e *log\1 - e / e *log\1 - e / e *log\1 - e /
- ------------------------- - ------------------------ + ------------------------- + ------------------------
3 3 3 3
$$\frac{e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{4 i \pi}{3}} \right)}}{3} - \frac{e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{4 i \pi}{3}} \right)}}{3} - \frac{e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{2 i \pi}{3}} \right)}}{3} + \frac{e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{2 i \pi}{3}} \right)}}{3}$$
-exp(-2*pi*i/3)*log(1 - exp_polar(2*pi*i/3))/3 - exp(2*pi*i/3)*log(1 - exp_polar(4*pi*i/3))/3 + exp(-2*pi*i/3)*log(1 - exp_polar(4*pi*i/3))/3 + exp(2*pi*i/3)*log(1 - exp_polar(2*pi*i/3))/3
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