Feb 21, 2025 · Well, the image equation is a different equation? One has $\frac1 {2024}$ on the right, and the other has $2024$ on the right?

I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx.$$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I ...

Nov 11, 2025 · I am evaluating the following integral: $$\\int_0^{1} \\left(\\tanh^{-1}(x) + \\tan^{-1}(x)\\right)^2 \\; dx$$ After using the Taylor series of the two functions, we ...

Jan 24, 2025 · Since the OP solve his/her problem, I just as well complete the solution: \begin {align} \frac {1} {n+1}\sum^n_ {k=1}\cos\left (\tfrac {k\pi} {2n}\right)&=\frac {n ...

Sep 11, 2024 · The following is a question from the Joint Entrance Examination (Main) from the 09 April 2024 evening shift: $$ \lim_ {x \to 0} \frac {e - (1 + 2x)^ {1/2x}} {x} $$ is equal to: (A) $0$ (B) $\frac { …

Jul 29, 2020 · Spherical Coordinate Homework Question Evaluate the triple integral of $f (x,y,z)=z (x^2+y^2+z^2)^ {−3/2}$ over the part of the ball $x^2+y^2+z^2\le 81$ defined by ...

Sep 13, 2016 · Compute:$$\prod_ {n=1}^ {\infty}\left (1+\frac {1} {2^n}\right)$$ I and my friend came across this product. Is the product till infinity equal to $1$? If no, what is the answer?

Sep 7, 2024 · The 1-D example is considerably easier, noticing that $\sum_ {n=1}^ {\lfloor x \rfloor} \frac {1} {n} = \ln (x) + \gamma + o (\frac {1} {x})$ The bounded function here is $\gamma$ and its average …

Oct 20, 2023 · You should instead multiply the limit in such way, that you get difference of cubes in the numerator. By multiplying with $\displaystyle \frac { (n^3 + n^2 + n + 1 ...

I'm supposed to calculate: $$\\lim_{n\\to\\infty} e^{-n} \\sum_{k=0}^{n} \\frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\\frac{1}{2 ...