Mister Exam
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How to use it?
- Sum of series:
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9/(9n^2+21n-8)
-
(3-sin*n)/n-lnn
-
1/(1+n^2)
-
e^(-n)
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Identical expressions
- three ^(n+ one)/factorial(n+ one)
- 3 to the power of (n plus 1) divide by factorial(n plus 1)
- three to the power of (n plus one) divide by factorial(n plus one)
- 3(n+1)/factorial(n+1)
- 3n+1/factorialn+1
- 3^n+1/factorialn+1
- 3^(n+1) divide by factorial(n+1)
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Similar expressions
- 3^(n-1)/factorial(n+1)
- 3^(n+1)/factorial(n-1)
Sum of series 3^(n+1)/factorial(n+1)
The solution
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oo ____ \ ` \ n + 1 \ 3 / -------- / (n + 1)! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{3^{n + 1}}{\left(n + 1\right)!}$$
Sum(3^(n + 1)/factorial(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3^{n + 1}}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{3^{n + 1}}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1} \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|\right)$$
Let's take the limit
we find
$$\frac{3^{n + 1}}{\left(n + 1\right)!}$$
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
$$a_{n} = \frac{3^{n + 1}}{\left(n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1} \left|{\frac{\left(n + 2\right)!}{\left(n + 1\right)!}}\right|\right)$$
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