Prove \(\displaystyle{\frac{{{1}}}{{{4}{n}^{{2}}-{1}}}}={\frac{{{1}}}{{{\left({2}{n}+{1}\right)}{\left({2}{n}-{1}\right)}}}}={\frac{{{1}}}{{{2}{\left({2}{n}-{1}\right)}}}}-{\frac{{{1}}}{{{2}{\left({2}{n}+{1}\right)}}}}\) Could you explain the operation in

How do you simplify ${\displaystyle (3x+\frac{1}{2})+(7x-4\frac{1}{2})}$?

How do you solve ${\displaystyle \frac{1}{3}x>6}$ ?

How do you write 0.4 as a fraction?

The average of a sequence of random variables does not converge to the mean of each variable in the sequence almost surely, where it is determined that the sequence of averages
$(S_n{)}_{n=2}^{\mathrm{\infty }},S_n=\frac{1}{n-1}\sum _{m=2}^n{X}_m$, $P(X_m=m)=P(X_m=-m)=\frac{1}{2m\log (m)},P(X_m=0)=1-\frac{1}{m\log (m)}$ does not converge almost surely to 0. But what do we know about the general behaviour of the said sequence $(S_n{)}_{=2}^{\mathrm{\infty }}$? How could we determine whether or not the sequence converges almost surely to any constant $c\in \mathbb{R}$?

Write an algebraic expression of the verbal expression. 20 divided by t to the fifth power.

I heard that one of the drawback of Riemann integral is;it is very laborious to find Riemann integral in higher dimensions e.g Calculating Riemann integral of bounded function $f:D\subset {\mathbb{R}}^{\mathbb{2}}\to \mathbb{R}$,where $D$ is compact domain of ${\mathbb{R}}^{\mathbb{2}}$.We can find it but it will require more work,and technically it is much difficult.As we know in lebesgue integration,we do partition of range rather than domain as we do in Riemann integration,and therefore lebesgue integration is believe to be easier to apply in case of higher dimensions rather than Riemann integral.

But to calculate integral of $f:D\subset {\mathbb{R}}^{\mathbb{2}}\to \mathbb{R}$ is same as double integral and I never saw any lebesgue approach to double or triple integral.

Can we actually find double integral via lebegue's technique of integration?Is this technique easier as observed by me?If it is so then why don't I see lebesgue's technique to calculate double integral

Proving that 31123319 is the largest number with a self-accounting property is 10213223. What is the largest such number?