Mister Exam
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- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
-
3/(n(n+2))
-
sin1/n
-
ln(n+1)/n
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((n+1)^n*3^(2n-1))/(2n+1)!
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Identical expressions
- (three *n+ one)/factorial(n)
- (3 multiply by n plus 1) divide by factorial(n)
- (three multiply by n plus one) divide by factorial(n)
- (3n+1)/factorial(n)
- 3n+1/factorialn
- (3*n+1) divide by factorial(n)
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Similar expressions
- ((-3^(n+1))/factorial(n+1))^n
- (3*n-1)/factorial(n)
- (n^2+3*n+1)/factorial(n+2)
- 3^(n+1)/factorial(n+1)
Sum of series (3*n+1)/factorial(n)
The solution
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oo ___ \ ` \ 3*n + 1 ) ------- / n! /__, n = 1
$$\sum_{n=1}^{\infty} \frac{3 n + 1}{n!}$$
Sum((3*n + 1)/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 n + 1}{n!}$$
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
$$a_{n} = \frac{3 n + 1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 1\right) \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3 n + 4}\right)$$
Let's take the limit
we find
$$\frac{3 n + 1}{n!}$$
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
$$a_{n} = \frac{3 n + 1}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 1\right) \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3 n + 4}\right)$$
Let's take the limit
we find
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