Mister Exam
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How to use it?
- Sum of series:
-
12/(n^2+4*n+3)
- x(1-x)^n
-
sqrt(3n+4)-sqrt(3n+1)
-
sin((2*n+1)/n^3)
-
Identical expressions
- sqrt(3n+ four)-sqrt(3n+ one)
- square root of (3n plus 4) minus square root of (3n plus 1)
- square root of (3n plus four) minus square root of (3n plus one)
- √(3n+4)-√(3n+1)
- sqrt3n+4-sqrt3n+1
-
Similar expressions
- sqrt(3n+4)+sqrt(3n+1)
- sqrt(3n+4)-sqrt(3n-1)
- sqrt(3n-4)-sqrt(3n+1)
Sum of series sqrt(3n+4)-sqrt(3n+1)
The solution
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oo ___ \ ` \ / _________ _________\ / \\/ 3*n + 4 - \/ 3*n + 1 / /__, n = 1
$$\sum_{n=1}^{\infty} \left(- \sqrt{3 n + 1} + \sqrt{3 n + 4}\right)$$
Sum(sqrt(3*n + 4) - sqrt(3*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \sqrt{3 n + 1} + \sqrt{3 n + 4}$$
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
$$a_{n} = - \sqrt{3 n + 1} + \sqrt{3 n + 4}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sqrt{3 n + 1} - \sqrt{3 n + 4}}{\sqrt{3 n + 4} - \sqrt{3 n + 7}}}\right|$$
Let's take the limit
we find
$$- \sqrt{3 n + 1} + \sqrt{3 n + 4}$$
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
$$a_{n} = - \sqrt{3 n + 1} + \sqrt{3 n + 4}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sqrt{3 n + 1} - \sqrt{3 n + 4}}{\sqrt{3 n + 4} - \sqrt{3 n + 7}}}\right|$$
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