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:
-
sin((n^2+3)/(n^3+2))^2
-
factorial(n+1)/((7^n*n))
-
n^(n+1)/factorial(n+1)
- ln(1+x/2)
-
Identical expressions
- thirty-four /(4n+ three)/(4n+ seven)
- 34 divide by (4n plus 3) divide by (4n plus 7)
- thirty minus four divide by (4n plus three) divide by (4n plus seven)
- 34/4n+3/4n+7
- 34 divide by (4n+3) divide by (4n+7)
-
Similar expressions
- 34/(4n+3)/(4n-7)
- 34/(4n-3)/(4n+7)
Sum of series 34/(4n+3)/(4n+7)
The solution
You have entered
[src]
oo ____ \ ` \ / 34 \ \ |-------| ) \4*n + 3/ / --------- / 4*n + 7 /___, n = -3
$$\sum_{n=-3}^{\infty} \frac{34 \frac{1}{4 n + 3}}{4 n + 7}$$
Sum((34/(4*n + 3))/(4*n + 7), (n, -3, oo))
The radius of convergence of the power series
Given number:
$$\frac{34 \frac{1}{4 n + 3}}{4 n + 7}$$
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
$$a_{n} = \frac{34}{\left(4 n + 3\right) \left(4 n + 7\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{4 n + 11}{4 n + 3}\right)$$
Let's take the limit
we find
$$\frac{34 \frac{1}{4 n + 3}}{4 n + 7}$$
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
$$a_{n} = \frac{34}{\left(4 n + 3\right) \left(4 n + 7\right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{4 n + 11}{4 n + 3}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
34*Gamma(-1/4) -------------- 45*Gamma(-5/4)
$$\frac{34 \Gamma\left(- \frac{1}{4}\right)}{45 \Gamma\left(- \frac{5}{4}\right)}$$
34*gamma(-1/4)/(45*gamma(-5/4))
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