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/((3n-2)(3n+1))
- x^n/sqrt(n+1)
-
12/(4n^2+12n+5)
- 1/(sqrt(k^4+7k^2+20))
-
Identical expressions
- twelve /(4n^ two +12n+ five)
- 12 divide by (4n squared plus 12n plus 5)
- twelve divide by (4n to the power of two plus 12n plus five)
- 12/(4n2+12n+5)
- 12/4n2+12n+5
- 12/(4n²+12n+5)
- 12/(4n to the power of 2+12n+5)
- 12/4n^2+12n+5
- 12 divide by (4n^2+12n+5)
-
Similar expressions
- 12/(4n^2-12n+5)
- 12/(4n^2+12n-5)
Sum of series 12/(4n^2+12n+5)
The solution
You have entered
[src]
oo ____ \ ` \ 12 \ --------------- / 2 / 4*n + 12*n + 5 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{12}{\left(4 n^{2} + 12 n\right) + 5}$$
Sum(12/(4*n^2 + 12*n + 5), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{12}{\left(4 n^{2} + 12 n\right) + 5}$$
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
$$a_{n} = \frac{12}{4 n^{2} + 12 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{12 \left(n + \frac{\left(n + 1\right)^{2}}{3} + \frac{17}{12}\right)}{4 n^{2} + 12 n + 5}\right)$$
Let's take the limit
we find
$$\frac{12}{\left(4 n^{2} + 12 n\right) + 5}$$
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
$$a_{n} = \frac{12}{4 n^{2} + 12 n + 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{12 \left(n + \frac{\left(n + 1\right)^{2}}{3} + \frac{17}{12}\right)}{4 n^{2} + 12 n + 5}\right)$$
Let's take the limit
we find
True
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