Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
3n
- nx^n!
-
(3-k)/(k*(k+3)*(k+1))
-
sqrt(n)sin(1/(2n))^2
-
Identical expressions
- n+ one /n* three ^n
- n plus 1 divide by n multiply by 3 to the power of n
- n plus one divide by n multiply by three to the power of n
- n+1/n*3n
- n+1/n3^n
- n+1/n3n
- n+1 divide by n*3^n
-
Similar expressions
- n-1/n*3^n
Sum of series n+1/n*3^n
The solution
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oo ____ \ ` \ / n\ \ | 3 | / |n + --| / \ n / /___, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{3^{n}}{n} + n\right)$$
Sum(n + 3^n/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3^{n}}{n} + n$$
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
$$a_{n} = \frac{3^{n}}{n} + n$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\frac{3^{n}}{n} + n}{\frac{3^{n + 1}}{n + 1} + n + 1}\right)$$
Let's take the limit
we find
$$\frac{3^{n}}{n} + n$$
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
$$a_{n} = \frac{3^{n}}{n} + n$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\frac{3^{n}}{n} + n}{\frac{3^{n + 1}}{n + 1} + n + 1}\right)$$
Let's take the limit
we find
False
False
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