Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
factorial(n)/(n^3+n+8)
-
1/((2n-1)*(2n+5))
-
-1/(2*n-1)!
- 3*(5*x-3)^n/((5^n*4))
-
Identical expressions
- - one /(two *n- one)!
- minus 1 divide by (2 multiply by n minus 1)!
- minus one divide by (two multiply by n minus one)!
- -1/(2n-1)!
- -1/2n-1!
- -1 divide by (2*n-1)!
-
Similar expressions
- (-1)^n*(2^(2n-1))/(2n-1)!
- (((-1)^(n+1))*(5^(n-1))*(x^(2n))-((-1)^(n+2))*(5^(2n-1))*(x^(2n-1)))/((2n-1)!)
- a^(2n-1)/(2n-1)!
- ((-1)^(n+1))*(pi^(2n-1))/((2n-1)!)
- 1/(2*n-1)!
- -1/(2*n+1)!
- (-1)/(2n-1)!
Sum of series -1/(2*n-1)!
The solution
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oo ___ \ ` \ -1 ) ---------- / (2*n - 1)! /__, n = 1
$$\sum_{n=1}^{\infty} - \frac{1}{\left(2 n - 1\right)!}$$
Sum(-1/factorial(2*n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{1}{\left(2 n - 1\right)!}$$
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
$$a_{n} = - \frac{1}{\left(2 n - 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 1\right)!}{\left(2 n - 1\right)!}}\right|$$
Let's take the limit
we find
$$- \frac{1}{\left(2 n - 1\right)!}$$
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
$$a_{n} = - \frac{1}{\left(2 n - 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 1\right)!}{\left(2 n - 1\right)!}}\right|$$
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