Mister Exam
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How to use it?
- Sum of series:
- sin(pi/2^n)
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sin(pi*n/3)
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1/(n+6)(n+7)
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(7n^2+5n-1)^3n+2/4n^2+2
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Identical expressions
- (7n^ two +5n- one)^3n+ two /4n^ two + two
- (7n squared plus 5n minus 1) cubed n plus 2 divide by 4n squared plus 2
- (7n to the power of two plus 5n minus one) cubed n plus two divide by 4n to the power of two plus two
- (7n2+5n-1)3n+2/4n2+2
- 7n2+5n-13n+2/4n2+2
- (7n²+5n-1)³n+2/4n²+2
- (7n to the power of 2+5n-1) to the power of 3n+2/4n to the power of 2+2
- 7n^2+5n-1^3n+2/4n^2+2
- (7n^2+5n-1)^3n+2 divide by 4n^2+2
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Similar expressions
- (7n^2+5n+1)^3n+2/4n^2+2
- (7n^2+5n-1)^3n+2/4n^2-2
- (7n^2-5n-1)^3n+2/4n^2+2
- (7n^2+5n-1)^3n-2/4n^2+2
Sum of series (7n^2+5n-1)^3n+2/4n^2+2
The solution
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oo ____ \ ` \ / 3 2 \ \ |/ 2 \ n | / |\7*n + 5*n - 1/ *n + -- + 2| / \ 2 / /___, n = 1
$$\sum_{n=1}^{\infty} \left(\left(\frac{n^{2}}{2} + n \left(\left(7 n^{2} + 5 n\right) - 1\right)^{3}\right) + 2\right)$$
Sum((7*n^2 + 5*n - 1)^3*n + n^2/2 + 2, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{n^{2}}{2} + n \left(\left(7 n^{2} + 5 n\right) - 1\right)^{3}\right) + 2$$
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
$$a_{n} = \frac{n^{2}}{2} + n \left(7 n^{2} + 5 n - 1\right)^{3} + 2$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{n^{2}}{2} + n \left(7 n^{2} + 5 n - 1\right)^{3} + 2}\right|}{\frac{\left(n + 1\right)^{2}}{2} + \left(n + 1\right) \left(5 n + 7 \left(n + 1\right)^{2} + 4\right)^{3} + 2}\right)$$
Let's take the limit
we find
$$\left(\frac{n^{2}}{2} + n \left(\left(7 n^{2} + 5 n\right) - 1\right)^{3}\right) + 2$$
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
$$a_{n} = \frac{n^{2}}{2} + n \left(7 n^{2} + 5 n - 1\right)^{3} + 2$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{n^{2}}{2} + n \left(7 n^{2} + 5 n - 1\right)^{3} + 2}\right|}{\frac{\left(n + 1\right)^{2}}{2} + \left(n + 1\right) \left(5 n + 7 \left(n + 1\right)^{2} + 4\right)^{3} + 2}\right)$$
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