Evaluating expressions color by number pdf. Is the product till infinity equal to $1$? If no, what is the answer? 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 {-2} { I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx. I see that the function has 2 Feb 21, 2025 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Sep 13, 2016 · Compute:$$\prod_ {n=1}^ {\infty}\left (1+\frac {1} {2^n}\right)$$ I and my friend came across this product. How would I go about evaluating this integral? $$\int_0^ {\infty}\frac {\ln (x^2+1)} {x^2+1}dx. Apr 23, 2017 · Evaluate the contour integral $\int_C\frac {z+1} {z^2-2z}dz$ using Cauchy's residue theorem, where $C$ is the circle $|z|=3$. $$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex p Mar 15, 2021 · Evaluating $\int_0^1 (1-x^2)^n dx$ [duplicate] Ask Question Asked 4 years, 7 months ago Modified 4 years, 7 months ago Evaluating $\int x \sqrt {x^2 - x}\ dx$ Ask Question Asked 2 years, 8 months ago Modified 10 months ago Evaluating $\int_ {0}^ {\pi}\ln (1+\cos x)\, dx$ [duplicate] Ask Question Asked 9 years, 10 months ago Modified 3 years, 4 months ago How would you evaluate the following series? $$\\lim_{n\\to\\infty} \\sum_{k=1}^{n^2} \\frac{n}{n^2+k^2} $$ Thanks. I see that the function has 2 . Is the product till infinity equal to $1$? If no, what is the answer? I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx. $$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I Nov 27, 2020 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. $$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex p Dec 7, 2018 · I see now how I can go about evaluating the limit itself although I still find the concept a little bit vague, as in considering a specific order for the expansion and then applying it for all the functions while leaving the rest of the series behind, it'll probably start making more sense in a few more examples. Jan 12, 2025 · Evaluating a Complex Integral involving Bessel Function Ask Question Asked 9 months ago Modified 9 months ago Feb 26, 2023 · I would like to solve the first problem of the 2023 MIT Integration Bee Finals, which is the following integral : $$\int_0^ {\pi/2} \frac {\sqrt [3] {\tan x}} { (\cos x + \sin x)^2}dx$$ I tried substitut How would you evaluate the following series? $$\\lim_{n\\to\\infty} \\sum_{k=1}^{n^2} \\frac{n}{n^2+k^2} $$ Thanks. Upvoting indicates when questions and answers are useful. Feb 21, 2025 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Dec 26, 2024 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. What's reputation and how do I get it? Instead, you can save this post to reference later. Feb 21, 2025 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. I see that the function has 2 Sep 13, 2016 · Compute:$$\prod_ {n=1}^ {\infty}\left (1+\frac {1} {2^n}\right)$$ I and my friend came across this product. uydac8p 6ca pu6lx 2xna vqgqh7f y68k9h jn00lr 5xowzuy cqp ukjw