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:
-
1/(n(n+2))
-
(2n-1)/2^n
-
sqrt(n)-2*sqrt(n+1)+sqrt(n+2)
-
log((n^2+1)/n^2)
-
Identical expressions
- ((n^ two +5n- three)/(4n+ seven))^n
- ((n squared plus 5n minus 3) divide by (4n plus 7)) to the power of n
- ((n to the power of two plus 5n minus three) divide by (4n plus seven)) to the power of n
- ((n2+5n-3)/(4n+7))n
- n2+5n-3/4n+7n
- ((n²+5n-3)/(4n+7))^n
- ((n to the power of 2+5n-3)/(4n+7)) to the power of n
- n^2+5n-3/4n+7^n
- ((n^2+5n-3) divide by (4n+7))^n
-
Similar expressions
- ((n^2-5n-3)/(4n+7))^n
- ((n^2+5n-3)/(4n-7))^n
- ((n^2+5n+3)/(4n+7))^n
Sum of series ((n^2+5n-3)/(4n+7))^n
The solution
You have entered
[src]
oo ____ \ ` \ n \ / 2 \ ) |n + 5*n - 3| / |------------| / \ 4*n + 7 / /___, n = 1
$$\sum_{n=1}^{\infty} \left(\frac{\left(n^{2} + 5 n\right) - 3}{4 n + 7}\right)^{n}$$
Sum(((n^2 + 5*n - 3)/(4*n + 7))^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{\left(n^{2} + 5 n\right) - 3}{4 n + 7}\right)^{n}$$
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
$$a_{n} = \left(\frac{n^{2} + 5 n - 3}{4 n + 7}\right)^{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{5 n + \left(n + 1\right)^{2} + 2}{4 n + 11}\right)^{- n - 1} \left|{\left(\frac{n^{2} + 5 n - 3}{4 n + 7}\right)^{n}}\right|\right)$$
Let's take the limit
we find
$$\left(\frac{\left(n^{2} + 5 n\right) - 3}{4 n + 7}\right)^{n}$$
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
$$a_{n} = \left(\frac{n^{2} + 5 n - 3}{4 n + 7}\right)^{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(\frac{5 n + \left(n + 1\right)^{2} + 2}{4 n + 11}\right)^{- n - 1} \left|{\left(\frac{n^{2} + 5 n - 3}{4 n + 7}\right)^{n}}\right|\right)$$
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