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Evaluating $\\prod_{n=1}^{\\infty}\\left(1+\\frac{1}{2^n}\\right)$. 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? Evaluating $ \\lim_{x \\to 0} \\frac{e - (1 + 2x)^{1/2x}}{x} $ without .... 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} {... Evaluating $\cos (i)$ - Mathematics Stack Exchange.
Evaluating $\\int_0^1 (1-x^2)^n dx$ - Mathematics Stack Exchange. Evaluating $\int_0^1 (1-x^2)^n dx$ [duplicate] Ask Question Asked 4 years, 8 months ago Modified 4 years, 7 months ago Evaluating $ \lim\limits_ {n\to\infty} \sum_ {k=1}^ {n^2} \frac {n} {n .... Furthermore, how would you evaluate the following series?
$$\\lim_{n\\to\\infty} \\sum_{k=1}^{n^2} \\frac{n}{n^2+k^2} $$ Thanks. contest math - Evaluating $4^ {5 \log_ {4\sqrt {2}} (3-\sqrt {6}) - 6 .... Here's another, seemingly monstrous question from a JEE Advanced preparation book. Evaluate the following expression: $$4^{5 \\log_{4\\sqrt{2}} (3-\\sqrt{6}) - 6\\log ... Evaluating $\\int_0^{\\infty}\\frac{\\ln(x^2+1)}{x^2+1}dx$. This perspective suggests that, how would I go about evaluating this integral?
$$\int_0^ {\infty}\frac {\ln (x^2+1)} {x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex p... calculus - Evaluating $\int \frac {1} { {x^4+1}} dx$ - Mathematics .... I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx.$$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I ... I'm supposed to calculate: $$\\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 ...
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