Mister Exam
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How to use it?
- Sum of series:
- x^n/n
-
1/(4n^2-1)
-
(1000^n)/n!
-
2n
-
Identical expressions
- (one thousand ^n)/n!
- (1000 to the power of n) divide by n!
- (one thousand to the power of n) divide by n!
- (1000n)/n!
- 1000n/n!
- 1000^n/n!
- (1000^n) divide by n!
Sum of series (1000^n)/n!
The solution
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oo ____ \ ` \ n \ 1000 / ----- / n! /___, n = 1
$$\sum_{n=1}^{\infty} \frac{1000^{n}}{n!}$$
Sum(1000^n/factorial(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1000^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = -1000$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-1000 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
$$\frac{1000^{n}}{n!}$$
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
$$a_{n} = \frac{1}{n!}$$
and
$$x_{0} = -1000$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-1000 + \lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|\right)$$
Let's take the limit
we find
$$R = \infty$$
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