Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
sin(pi/n)
- x^2/n^3
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sqrt(n+2)-2*sqrt(n-1)+sqrt(n)
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nlogn
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Identical expressions
- sqrt(n+ two)- two *sqrt(n- one)+sqrt(n)
- square root of (n plus 2) minus 2 multiply by square root of (n minus 1) plus square root of (n)
- square root of (n plus two) minus two multiply by square root of (n minus one) plus square root of (n)
- √(n+2)-2*√(n-1)+√(n)
- sqrt(n+2)-2sqrt(n-1)+sqrt(n)
- sqrtn+2-2sqrtn-1+sqrtn
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Similar expressions
- sqrt(n+2)-2*sqrt(n+1)+sqrt(n)
- sqrt(n-2)-2*sqrt(n-1)+sqrt(n)
- sqrt(n+2)+2*sqrt(n-1)+sqrt(n)
- sqrt(n+2)-(2*sqrt(n-1))+sqrt(n)
- sqrt(n+2)-2*sqrt(n-1)-sqrt(n)
Sum of series sqrt(n+2)-2*sqrt(n-1)+sqrt(n)
The solution
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oo ___ \ ` \ / _______ _______ ___\ / \\/ n + 2 - 2*\/ n - 1 + \/ n / /__, n = 1
$$\sum_{n=1}^{\infty} \left(\sqrt{n} + \left(- 2 \sqrt{n - 1} + \sqrt{n + 2}\right)\right)$$
Sum(sqrt(n + 2) - 2*sqrt(n - 1) + sqrt(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sqrt{n} + \left(- 2 \sqrt{n - 1} + \sqrt{n + 2}\right)$$
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
$$a_{n} = \sqrt{n} - 2 \sqrt{n - 1} + \sqrt{n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sqrt{n} - 2 \sqrt{n - 1} + \sqrt{n + 2}}{- 2 \sqrt{n} + \sqrt{n + 1} + \sqrt{n + 3}}}\right|$$
Let's take the limit
we find
$$\sqrt{n} + \left(- 2 \sqrt{n - 1} + \sqrt{n + 2}\right)$$
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
$$a_{n} = \sqrt{n} - 2 \sqrt{n - 1} + \sqrt{n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sqrt{n} - 2 \sqrt{n - 1} + \sqrt{n + 2}}{- 2 \sqrt{n} + \sqrt{n + 1} + \sqrt{n + 3}}}\right|$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
Numerical answer
The series diverges
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