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:
-
sqrt((n+4)/((n^4)+4))
-
(7^n-3^n)/21^n
- cos(kx)
- ((-1)^(n-1))*(5^n+2)*x^n
-
Identical expressions
- (four *(- one)^(n+ one))/(seven ^n)
- (4 multiply by ( minus 1) to the power of (n plus 1)) divide by (7 to the power of n)
- (four multiply by ( minus one) to the power of (n plus one)) divide by (seven to the power of n)
- (4*(-1)(n+1))/(7n)
- 4*-1n+1/7n
- (4(-1)^(n+1))/(7^n)
- (4(-1)(n+1))/(7n)
- 4-1n+1/7n
- 4-1^n+1/7^n
- (4*(-1)^(n+1)) divide by (7^n)
-
Similar expressions
- (4*(-1)^(n-1))/(7^n)
- (4*(1)^(n+1))/(7^n)
Sum of series (4*(-1)^(n+1))/(7^n)
The solution
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oo ____ \ ` \ n + 1 \ 4*(-1) ) ----------- / n / 7 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{4 \left(-1\right)^{n + 1}}{7^{n}}$$
Sum((4*(-1)^(n + 1))/7^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{4 \left(-1\right)^{n + 1}}{7^{n}}$$
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
$$a_{n} = 4 \left(-1\right)^{n + 1}$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{4 \left(-1\right)^{n + 1}}{7^{n}}$$
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
$$a_{n} = 4 \left(-1\right)^{n + 1}$$
and
$$x_{0} = -7$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-7 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
False
$$R = 0$$
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