Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
-
How to use it?
- Sum of series:
-
(1+n)/(1+n^2)
-
1/(n*ln(n+1))
- ((n^3+3n+1)^(1/2))arcsin(2/n^3)
-
3/((n+2)^2-1)
-
Identical expressions
- three /((n+ two)^ two - one)
- 3 divide by ((n plus 2) squared minus 1)
- three divide by ((n plus two) to the power of two minus one)
- 3/((n+2)2-1)
- 3/n+22-1
- 3/((n+2)²-1)
- 3/((n+2) to the power of 2-1)
- 3/n+2^2-1
- 3 divide by ((n+2)^2-1)
-
Similar expressions
- 3/((n+2)^2+1)
- 3/((n-2)^2-1)
Sum of series 3/((n+2)^2-1)
The solution
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oo ____ \ ` \ 3 \ ------------ / 2 / (n + 2) - 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{3}{\left(n + 2\right)^{2} - 1}$$
Sum(3/((n + 2)^2 - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3}{\left(n + 2\right)^{2} - 1}$$
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
$$a_{n} = \frac{3}{\left(n + 2\right)^{2} - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{3 \left(\frac{\left(n + 3\right)^{2}}{3} - \frac{1}{3}\right)}{\left(n + 2\right)^{2} - 1}\right)$$
Let's take the limit
we find
$$\frac{3}{\left(n + 2\right)^{2} - 1}$$
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
$$a_{n} = \frac{3}{\left(n + 2\right)^{2} - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{3 \left(\frac{\left(n + 3\right)^{2}}{3} - \frac{1}{3}\right)}{\left(n + 2\right)^{2} - 1}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
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