Mister Exam
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- Product of series
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How to use it?
- Sum of series:
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1/n*(n+1)
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7+k
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6/(3n-1)(3n+5)
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6/((3*n-1)*(3*n+5))
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Identical expressions
- one / four + one / twenty-eight + one / seventy + one /(3n- two)*(3n+ one)
- 1 divide by 4 plus 1 divide by 28 plus 1 divide by 70 plus 1 divide by (3n minus 2) multiply by (3n plus 1)
- one divide by four plus one divide by twenty minus eight plus one divide by seventy plus one divide by (3n minus two) multiply by (3n plus one)
- 1/4+1/28+1/70+1/(3n-2)(3n+1)
- 1/4+1/28+1/70+1/3n-23n+1
- 1 divide by 4+1 divide by 28+1 divide by 70+1 divide by (3n-2)*(3n+1)
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Similar expressions
- 1/4+1/28-1/70+1/(3n-2)*(3n+1)
- 1/4+1/28+1/70-1/(3n-2)*(3n+1)
- 1/4+1/28+1/70+1/(3n-2)*(3n-1)
- 1/4-1/28+1/70+1/(3n-2)*(3n+1)
- (1/4)+(1/28)+(1/70)+(1/(3n-2)(3n+1))
- 1/4+1/28+1/70+1/(3n+2)*(3n+1)
Sum of series 1/4+1/28+1/70+1/(3n-2)*(3n+1)
The solution
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oo ___ \ ` \ / 3*n + 1\ ) |1/4 + 1/28 + 1/70 + -------| / \ 3*n - 2/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(\left(\frac{1}{70} + \left(\frac{1}{28} + \frac{1}{4}\right)\right) + \frac{3 n + 1}{3 n - 2}\right)$$
Sum(1/4 + 1/28 + 1/70 + (3*n + 1)/(3*n - 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{70} + \left(\frac{1}{28} + \frac{1}{4}\right)\right) + \frac{3 n + 1}{3 n - 2}$$
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
$$a_{n} = \frac{3}{10} + \frac{3 n + 1}{3 n - 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{3}{10} + \frac{3 n + 1}{3 n - 2}}\right|}{\frac{3}{10} + \frac{3 n + 4}{3 n + 1}}\right)$$
Let's take the limit
we find
$$\left(\frac{1}{70} + \left(\frac{1}{28} + \frac{1}{4}\right)\right) + \frac{3 n + 1}{3 n - 2}$$
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
$$a_{n} = \frac{3}{10} + \frac{3 n + 1}{3 n - 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{3}{10} + \frac{3 n + 1}{3 n - 2}}\right|}{\frac{3}{10} + \frac{3 n + 4}{3 n + 1}}\right)$$
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