Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
-
2^n/n^2
-
5^(n+2)/(4^(n-1)*7^n)
-
((n^2)/(9n^2-1))^2
-
sin(n)/2^n
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Identical expressions
- (one /n)^(n- three)
- (1 divide by n) to the power of (n minus 3)
- (one divide by n) to the power of (n minus three)
- (1/n)(n-3)
- 1/nn-3
- 1/n^n-3
- (1 divide by n)^(n-3)
-
Similar expressions
- (1/n)^(n+3)
Sum of series (1/n)^(n-3)
The solution
You have entered
[src]
oo ____ \ ` \ n - 3 \ /1\ / |-| / \n/ /___, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{1}{n}\right)^{n - 3}$$
Sum((1/n)^(n - 3), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{n}\right)^{n - 3}$$
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
$$a_{n} = \left(\frac{1}{n}\right)^{n - 3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(n^{3 - n} \left(n + 1\right)^{n - 2}\right)$$
Let's take the limit
we find
$$\left(\frac{1}{n}\right)^{n - 3}$$
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
$$a_{n} = \left(\frac{1}{n}\right)^{n - 3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(n^{3 - n} \left(n + 1\right)^{n - 2}\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
[src]
oo ___ \ ` \ 3 - n / n /__, n = 1
$$\sum_{n=1}^{\infty} n^{3 - n}$$
Sum(n^(3 - n), (n, 1, oo))
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