Mister Exam
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- Product of series
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How to use it?
- Sum of series:
-
1/7n
-
1/(n*ln^2n)
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1/factorial(n^1)
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1+0,1+0,01+0,001+0,0001
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Identical expressions
- one + zero , one + zero , one + zero , one + zero , one
- 1 plus 0,1 plus 0,01 plus 0,001 plus 0,0001
- one plus zero , one plus zero , one plus zero , one plus zero , one
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Similar expressions
- 1+0,1+0,01+0,001-0,0001
- 1+0,1-0,01+0,001+0,0001
- 1+0,1+0,01-0,001+0,0001
- 1-0,1+0,01+0,001+0,0001
Sum of series 1+0,1+0,01+0,001+0,0001
The solution
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oo __ \ ` ) (1/10 + 1 + 1/100 + 1/1000 + 0.0001) /_, n = 1
$$\sum_{n=1}^{\infty} \left(0.0001 + \left(\frac{1}{1000} + \left(\frac{1}{100} + \left(\frac{1}{10} + 1\right)\right)\right)\right)$$
Sum(1/10 + 1 + 1/100 + 1/1000 + 0.0001, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$0.0001 + \left(\frac{1}{1000} + \left(\frac{1}{100} + \left(\frac{1}{10} + 1\right)\right)\right)$$
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
$$a_{n} = 1.1111$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$0.0001 + \left(\frac{1}{1000} + \left(\frac{1}{100} + \left(\frac{1}{10} + 1\right)\right)\right)$$
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
$$a_{n} = 1.1111$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
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