Mister Exam
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- Taylor Series Step by Step
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- Product of series
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How to use it?
- Sum of series:
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(n-1)/2^(n+1)
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6/(n^2-10n+24)
- a^n
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2^(2*n-1)/9^n
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Identical expressions
- one + one /x+ one /x^ two ++ one /x^ three + one /x^ four + one /x^ five + one /x^ six + one /x^ seven + one /x^ eight + one /x^ nine + one /x^ ten
- 1 plus 1 divide by x plus 1 divide by x squared plus plus 1 divide by x cubed plus 1 divide by x to the power of 4 plus 1 divide by x to the power of 5 plus 1 divide by x to the power of 6 plus 1 divide by x to the power of 7 plus 1 divide by x to the power of 8 plus 1 divide by x to the power of 9 plus 1 divide by x to the power of 10
- one plus one divide by x plus one divide by x to the power of two plus plus one divide by x to the power of three plus one divide by x to the power of four plus one divide by x to the power of five plus one divide by x to the power of six plus one divide by x to the power of seven plus one divide by x to the power of eight plus one divide by x to the power of nine plus one divide by x to the power of ten
- 1+1/x+1/x2++1/x3+1/x4+1/x5+1/x6+1/x7+1/x8+1/x9+1/x10
- 1+1/x+1/x²++1/x³+1/x⁴+1/x⁵+1/x⁶+1/x⁷+1/x⁸+1/x⁹+1/x^10
- 1+1/x+1/x to the power of 2++1/x to the power of 3+1/x to the power of 4+1/x to the power of 5+1/x to the power of 6+1/x to the power of 7+1/x to the power of 8+1/x to the power of 9+1/x to the power of 10
- 1+1 divide by x+1 divide by x^2++1 divide by x^3+1 divide by x^4+1 divide by x^5+1 divide by x^6+1 divide by x^7+1 divide by x^8+1 divide by x^9+1 divide by x^10
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Similar expressions
- 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6-1/x^7+1/x^8+1/x^9+1/x^10
- 1-1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5-1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7-1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8-1/x^9+1/x^10
- 1+1/x+1/x^2++1/x^3-1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2-+1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9-1/x^10
- 1+1/x+1/x^2++1/x^3+1/x^4-1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x+1/x^2+-1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
- 1+1/x-1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
Sum of series 1+1/x+1/x^2++1/x^3+1/x^4+1/x^5+1/x^6+1/x^7+1/x^8+1/x^9+1/x^10
The solution
You have entered
[src]
oo ____ \ ` \ / 1 1 1 1 1 1 1 1 1 1 \ \ |1 + - + -- + -- + -- + -- + -- + -- + -- + -- + ---| / | x 2 3 4 5 6 7 8 9 10| / \ x x x x x x x x x / /___, x = 2
$$\sum_{x=2}^{\infty} \left(\left(\left(\left(\left(\left(\left(\left(\left(\left(1 + \frac{1}{x}\right) + \frac{1}{x^{2}}\right) + \frac{1}{x^{3}}\right) + \frac{1}{x^{4}}\right) + \frac{1}{x^{5}}\right) + \frac{1}{x^{6}}\right) + \frac{1}{x^{7}}\right) + \frac{1}{x^{8}}\right) + \frac{1}{x^{9}}\right) + \frac{1}{x^{10}}\right)$$
Sum(1 + 1/x + 1/(x^2) + 1/(x^3) + 1/(x^4) + 1/(x^5) + 1/(x^6) + 1/(x^7) + 1/(x^8) + 1/(x^9) + 1/(x^10), (x, 2, oo))
The radius of convergence of the power series
Given number:
$$\left(\left(\left(\left(\left(\left(\left(\left(\left(1 + \frac{1}{x}\right) + \frac{1}{x^{2}}\right) + \frac{1}{x^{3}}\right) + \frac{1}{x^{4}}\right) + \frac{1}{x^{5}}\right) + \frac{1}{x^{6}}\right) + \frac{1}{x^{7}}\right) + \frac{1}{x^{8}}\right) + \frac{1}{x^{9}}\right) + \frac{1}{x^{10}}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = 1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}} + \frac{1}{x^{5}} + \frac{1}{x^{6}} + \frac{1}{x^{7}} + \frac{1}{x^{8}} + \frac{1}{x^{9}} + \frac{1}{x^{10}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}} + \frac{1}{x^{5}} + \frac{1}{x^{6}} + \frac{1}{x^{7}} + \frac{1}{x^{8}} + \frac{1}{x^{9}} + \frac{1}{x^{10}}}{1 + \frac{1}{x + 1} + \frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x + 1\right)^{3}} + \frac{1}{\left(x + 1\right)^{4}} + \frac{1}{\left(x + 1\right)^{5}} + \frac{1}{\left(x + 1\right)^{6}} + \frac{1}{\left(x + 1\right)^{7}} + \frac{1}{\left(x + 1\right)^{8}} + \frac{1}{\left(x + 1\right)^{9}} + \frac{1}{\left(x + 1\right)^{10}}}\right)$$
Let's take the limit
we find
$$\left(\left(\left(\left(\left(\left(\left(\left(\left(1 + \frac{1}{x}\right) + \frac{1}{x^{2}}\right) + \frac{1}{x^{3}}\right) + \frac{1}{x^{4}}\right) + \frac{1}{x^{5}}\right) + \frac{1}{x^{6}}\right) + \frac{1}{x^{7}}\right) + \frac{1}{x^{8}}\right) + \frac{1}{x^{9}}\right) + \frac{1}{x^{10}}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = 1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}} + \frac{1}{x^{5}} + \frac{1}{x^{6}} + \frac{1}{x^{7}} + \frac{1}{x^{8}} + \frac{1}{x^{9}} + \frac{1}{x^{10}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{x \to \infty}\left(\frac{1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + \frac{1}{x^{4}} + \frac{1}{x^{5}} + \frac{1}{x^{6}} + \frac{1}{x^{7}} + \frac{1}{x^{8}} + \frac{1}{x^{9}} + \frac{1}{x^{10}}}{1 + \frac{1}{x + 1} + \frac{1}{\left(x + 1\right)^{2}} + \frac{1}{\left(x + 1\right)^{3}} + \frac{1}{\left(x + 1\right)^{4}} + \frac{1}{\left(x + 1\right)^{5}} + \frac{1}{\left(x + 1\right)^{6}} + \frac{1}{\left(x + 1\right)^{7}} + \frac{1}{\left(x + 1\right)^{8}} + \frac{1}{\left(x + 1\right)^{9}} + \frac{1}{\left(x + 1\right)^{10}}}\right)$$
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