Mister Exam
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How to use it?
- Sum of series:
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2^n/n^2
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24/(9n^2-12n-5)
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n^(2/3)*arctg(1/n^2)
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2^(2*n-1)/9^n
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Identical expressions
- (- one)^n+ one /(2n+ one)!
- ( minus 1) to the power of n plus 1 divide by (2n plus 1)!
- ( minus one) to the power of n plus one divide by (2n plus one)!
- (-1)n+1/(2n+1)!
- -1n+1/2n+1!
- -1^n+1/2n+1!
- (-1)^n+1 divide by (2n+1)!
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Similar expressions
- (-1)^n+1/(2n-1)!
- (-1)^n-1/(2n+1)!
- (1)^n+1/(2n+1)!
Sum of series (-1)^n+1/(2n+1)!
The solution
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oo ___ \ ` \ / n 1 \ ) |(-1) + ----------| / \ (2*n + 1)!/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(\left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}\right)$$
Sum((-1)^n + 1/factorial(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}}{\left(-1\right)^{n + 1} + \frac{1}{\left(2 n + 3\right)!}}}\right|$$
Let's take the limit
we find
$$\left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}}{\left(-1\right)^{n + 1} + \frac{1}{\left(2 n + 3\right)!}}}\right|$$
Let's take the limit
we find
True
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