Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
- t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
- (-1)^nx^n/n!
-
5/(25*n^2-5*n-6)
- ((3*x)^n)/n!
-
Identical expressions
- ((three *x)^n)/n!
- ((3 multiply by x) to the power of n) divide by n!
- ((three multiply by x) to the power of n) divide by n!
- ((3*x)n)/n!
- 3*xn/n!
- ((3x)^n)/n!
- ((3x)n)/n!
- 3xn/n!
- 3x^n/n!
- ((3*x)^n) divide by n!
Sum of series ((3*x)^n)/n!
The solution
You have entered
[src]
oo ____ \ ` \ n \ (3*x) / ------ / n! /___, n = 2
$$\sum_{n=2}^{\infty} \frac{\left(3 x\right)^{n}}{n!}$$
Sum((3*x)^n/factorial(n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(3 x\right)^{n}}{n!}$$
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
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 3$$
then
$$R = \frac{\lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3}$$
Let's take the limit
we find
$$R = \infty$$
$$\frac{\left(3 x\right)^{n}}{n!}$$
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
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 3$$
then
$$R = \frac{\lim_{n \to \infty} \left|{\frac{\left(n + 1\right)!}{n!}}\right|}{3}$$
Let's take the limit
we find
$$R = \infty$$
The answer
[src]
/ 2 2*x \
|- - - --- 3*x|
2 | 9 3 2*e |
9*x *|--------- + ------|
| 2 2 |
\ x 9*x /
-------------------------
2
$$\frac{9 x^{2} \left(\frac{- \frac{2 x}{3} - \frac{2}{9}}{x^{2}} + \frac{2 e^{3 x}}{9 x^{2}}\right)}{2}$$
9*x^2*((-2/9 - 2*x/3)/x^2 + 2*exp(3*x)/(9*x^2))/2
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