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:
- (-1)^nx^n/n!
- ((2*n+4)/(5*n+7))^n*x^n
-
(n-2^(3*n))/(n+1)^(1/3)
- (-1)^(2n+1)*x^(2n+1)/(2n+1)!
-
Identical expressions
- (two n+ one)/2^n
- (2n plus 1) divide by 2 to the power of n
- (two n plus one) divide by 2 to the power of n
- (2n+1)/2n
- 2n+1/2n
- 2n+1/2^n
- (2n+1) divide by 2^n
-
Similar expressions
- (2n-1)/2^n
Sum of series (2n+1)/2^n
The solution
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oo ____ \ ` \ 2*n + 1 \ ------- / n / 2 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{2 n + 1}{2^{n}}$$
Sum((2*n + 1)/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{2 n + 1}{2^{n}}$$
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
$$a_{n} = 2 n + 1$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{2 n + 1}{2 n + 3}\right)\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{2 n + 1}{2^{n}}$$
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
$$a_{n} = 2 n + 1$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(\frac{2 n + 1}{2 n + 3}\right)\right)$$
Let's take the limit
we find
False
$$R = 0$$
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