limit of sin^-1(xy-2)/tan^-1(3xy-6) as (x y) approaches to (2 1)

August 20, 2021Posted inAdvanced Math, Mathematics, Multivariable Calculus, Real Analysis, Trigonometry
Question : Find the limit of sin^-1(xy-2)/tan^-1(3xy-6) as (x y) approaches to (2 1) \[{\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{\tan^{-1}{\left(3xy-6\right)}}} \] Solution : We shall let the limit is \[ L={\rm limit}_{\left(x,y\right)\rightarrow(2,1)}{\frac{\sin^{-1}{\left(xy-2\right)}}{\tan^{-1}{\left(3xy-6\right)}}}\] Multiplying and…

Solved Sum of the series 1/1.2-1/2.3+1/3.4-1/4.5+…

August 19, 2021Posted inAbstract Algebra, Advanced Math, Multivariable Calculus, Number Theory, Real Analysis, Trigonometry
Question : Sum of the series 1/1.2-1/2.3+1/3.4-1/4.5+… Solution : Let the Sum of the series is \[S=\ \frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\frac{1}{4.5}+\ldots..\] \[=\left(\frac{1}{1}-\frac{1}{2}\right)-\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)-\left(\frac{1}{4}-\frac{1}{5}\right)+\ldots.\] \[=\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots.\right)+\left(-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots..\right)\] \[=\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots.\right)+\left(-1+1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots..\right)\] We shall use formula of the series \[\log{\left(1+x\right)}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots..\] where…