Mister Exam
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- Product of series
-
How to use it?
- Sum of series:
-
(-1)^nsin(1/n)
-
1/(n*(n+1)*(n+2))
-
1/n^4
-
factorial(n)*3^n/n^n
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Identical expressions
- (- one)^nsin(one /n)
- ( minus 1) to the power of n sinus of (1 divide by n)
- ( minus one) to the power of n sinus of (one divide by n)
- (-1)nsin(1/n)
- -1nsin1/n
- -1^nsin1/n
- (-1)^nsin(1 divide by n)
-
Similar expressions
- (1)^nsin(1/n)
Sum of series (-1)^nsin(1/n)
The solution
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oo ___ \ ` \ n /1\ ) (-1) *sin|-| / \n/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(-1\right)^{n} \sin{\left(\frac{1}{n} \right)}$$
Sum((-1)^n*sin(1/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(-1\right)^{n} \sin{\left(\frac{1}{n} \right)}$$
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
$$a_{n} = \sin{\left(\frac{1}{n} \right)}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{1}{n} \right)}}{\sin{\left(\frac{1}{n + 1} \right)}}}\right|\right)$$
Let's take the limit
we find
$$\left(-1\right)^{n} \sin{\left(\frac{1}{n} \right)}$$
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
$$a_{n} = \sin{\left(\frac{1}{n} \right)}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty} \left|{\frac{\sin{\left(\frac{1}{n} \right)}}{\sin{\left(\frac{1}{n + 1} \right)}}}\right|\right)$$
Let's take the limit
we find
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