If x^a= y, y^b=z and z^c=x then prove that abc = 1
Answers
Answer:
Meaning of the expression
Here is the explanation for each symbol.
\displaystyle\red{\bullet}\ \sum^{n}_{k=1}\dfrac{1}{k}∙
k=1
∑
n
k
1
is the area of the rectangles.
\displaystyle\red{\bullet}\ \ln n∙ lnn is the area bounded by y=\dfrac{1}{x}y=
x
1
, x=1x=1 , x=nx=n and the x-axis.
\displaystyle\red{\bullet}\ \lim_{x\to\infty}∙
x→∞
lim
is the approaching value of an expression, as calculation repeats more and more times.
In the attachment, three areas are bounded by different colors.
\displaystyle\red{\bullet}\ \text{Grey region: The area that the domain of }\ln n\text{ cannot cover.}∙ Grey region: The area that the domain of lnn cannot cover.
\displaystyle\red{\bullet}\ \text{Red region: The area under }y=\dfrac{1}{x}\text{.}∙ Red region: The area under y=
x
1
.
\displaystyle\red{\bullet}\ \text{Purple region: The difference between the curve and rectangle.}∙ Purple region: The difference between the curve and rectangle.
\large\underline{\text{Note}}
Note
The number, \displaystyle\gamma=\lim_{x\to\infty}(\sum^{n}_{k=1}\dfrac{1}{k}-\ln n)γ=
x→∞
lim
(
k=1
∑
n
k
1
−lnn) is Euler-Mascheroni constant.
\large\underline{\text{Rational or irrational?}}
Rational or irrational?
It is not found whether the value is rational or irrational. By continued fraction method, it is proved that the denominator must be greater than 10^{244663}10
244663
by Papanikolaou in 1997.
\large\underline{\text{Graphical property}}
Graphical property
The number has the following property about the area.
\red{\bullet}\ \gamma=\text{(Area under the curve }\dfrac{1}{x})-\text{(Harmonic series)}∙ γ=(Area under the curve
x
1
)−(Harmonic series)
This property is used in the approximation of the harmonic series, as \gammaγ is around 0.577. And it is since the area of the purple region gets less and less. We get the following approximation.
\red{\bullet}\ \ln n-\gamma\approx\text{(Harmonic series)}∙ lnn−γ≈(Harmonic series)
Step-by-step explanation:
which is required solution of this question
