Mister Exam
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- Taylor Series Step by Step
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- Product of series
-
How to use it?
- Sum of series:
-
(arcsin(1/n))^n
-
3n
-
ln(n/n+1)
-
factorial(1000)/(factorial(450)*factorial(1000-450))
-
Identical expressions
- (- one)^n(x^(14n)/n!)
- ( minus 1) to the power of n(x to the power of (14n) divide by n!)
- ( minus one) to the power of n(x to the power of (14n) divide by n!)
- (-1)n(x(14n)/n!)
- -1nx14n/n!
- -1^nx^14n/n!
- (-1)^n(x^(14n) divide by n!)
-
Similar expressions
- (1)^n(x^(14n)/n!)
Sum of series (-1)^n(x^(14n)/n!)
The solution
You have entered
[src]
oo ____ \ ` \ 14*n \ n x / (-1) *----- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \left(-1\right)^{n} \frac{x^{14 n}}{n!}$$
Sum((-1)^n*(x^(14*n)/factorial(n)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(-1\right)^{n} \frac{x^{14 n}}{n!}$$
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
$$a_{n} = \frac{\left(-1\right)^{n}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 14$$
,
$$c = 1$$
then
$$R^{14} = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R^{14} = \infty$$
$$R = \infty$$
$$\left(-1\right)^{n} \frac{x^{14 n}}{n!}$$
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
$$a_{n} = \frac{\left(-1\right)^{n}}{n!}$$
and
$$x_{0} = 0$$
,
$$d = 14$$
,
$$c = 1$$
then
$$R^{14} = \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|$$
Let's take the limit
we find
$$R^{14} = \infty$$
$$R = \infty$$
The answer
[src]
/ 14\
| -x |
14 | 1 e |
-x *|--- - -----|
| 14 14 |
\x x /
$$- x^{14} \left(\frac{1}{x^{14}} - \frac{e^{- x^{14}}}{x^{14}}\right)$$
-x^14*(x^(-14) - exp(-x^14)/x^14)
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