I'm having trouble understanding how ML estimates were used to find the upper bound of the Contour Integral defined in $(2.)$. My understanding of the example detailing the integral in question can be followed through from $(1).$ $(0.0)$ If $f$ is a complex-valued, continuous function on the co...

I am wondering if there is a way to solve this definite integral depicted below: $$\int_0^1 \log(-\log(x)\sin(x))\text{d}x.$$ Mathematica numerically evaluates this integral to $-1.63394$. I am not experienced with these types of integrals and have tried approaches in related questions with n...

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ $\bf{My\; Try::}$ Using $$\left(\sqrt{k}+\sqrt{k-1}\right)<2\sqrt{k}<\left(\sqrt{k+1}+\sqrt{k}\ri...

I'm having trouble understanding how ML estimates were used to find the upper bound of the Contour Integral defined in $(2.)$. My understanding of the example detailing the integral in question can be followed through from $(1).$ $(0.0)$ If $f$ is a complex-valued, continuous function on the co...

Are users encouraged to ask simple math questions in this forum as well? Most of the questions here seem advanced for a math learner.

Notice that: $$60.41<60.5<61.24$$ but, $$\lfloor60.5\rfloor=60\ne61$$ so your logic is wrong. To correct it, simply note that: $$\sum_{k=2}^{1000}\frac1{\sqrt k}=\sum_{a=2}^{10}\frac1{\sqrt a}+\sum_{b=11}^{1000}\frac1{\sqrt b}$$ Likewise, you will find that $$\sum_{a=2}^{10}\frac1{\sqrt a...