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  1. J

    Binomial Ratio

    Evaluation of the sum \displaystyle \frac{\sum^{r}_{k=0}\binom{n}{2k}\cdot \binom{n-2k}{r-k}}{\sum^{n}_{k=r}\binom{n}{k}\cdot \binom{2k}{2r}(0.75)^{n-k}\cdot (0.5)^{2k-2r}}\ \ ,\ \ (n\geq 2r)
  2. J

    Determinant of matrix of order 0*0

    A have a doubt : If \displaystyle A=[a_{ij}]_{0\times 0} is a matrix . Then \displaystyle |A| =
  3. J

    Exponential and factorial inequality

    Thanks Tywebb. I have tried like this way I am assuming \displaystyle n=2k, k\geq 3, k\in\mathbb{Z} So we have \displaystyle \frac{n!}{\bigg(\frac{n}{2}\bigg)^n}=2\prod^{\frac{n}{2}-1}_{r=1}\frac{\bigg(\frac{n}{2}-r\bigg)\bigg(\frac{n}{2}+r\bigg)}{\frac{n}{2}\cdot...
  4. J

    Exponential and factorial inequality

    Thanks cossine. Using your hint , We use Stirling approximation \displaystyle n!\approx \bigg(\frac{n}{e}\bigg)^{n}\sqrt{2\pi n}\approx \bigg(\frac{n}{e}\bigg)^n. So we have \displaystyle \frac{n}{\bigg(n!\bigg)^{\frac{1}{n}}}\approx e>2\Longrightarrow n!<\bigg(\frac{n}{2}\bigg)^n
  5. J

    Exponential and factorial inequality

    Proving the result \displaystyle \frac{n^n}{n!}>2^n, n>6
  6. J

    limit with consecutive factorials

    Thanks Tywebb.
  7. J

    limit with consecutive factorials

    Thanks Tywebb. Can we solve it without using Stolz and Stirling approximation like definite integration If yes then please explain me.
  8. J

    limit with consecutive factorials

    Evaluation of \displaystyle \lim_{n\rightarrow \infty}\bigg(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}\bigg)
  9. J

    binomial limit

    Thanks Tywebb for different Methods.
  10. J

    binomial limit

    Evaluation of \displaystyle \lim_{n\rightarrow \infty}\bigg[\binom{n}{0}\cdot \binom{n}{1}\cdot \binom{n}{2}\cdots \cdots \binom{n}{n}\bigg]^{\frac{1}{n(n+1)}}
  11. J

    Irrational Integration

    (a)\; $Evaluation of $\int\frac{\sqrt{9x^2+4x+6}}{8x^2+4x+6}dx$ (b)\; $Evaluation of $\int\frac{x^2(x\sec x+\tan x)}{(x\tan x-1)^2}dx.$
  12. J

    floor summ

    $Consider a sequence $b_{n}$ given as $b_{1}=\frac{1}{3}$ $ and $b_{n+1}=b^2_{n}+b_{n}$. Then value of $\lfloor \sum^{2008}_{k=2}\frac{1}{b_{k}}\rfloor.
  13. J

    Definite Integration

    $Evaluation of $\int^{\frac{\pi}{2}}_{0}\cos x\cdot \ln(\cos x)dx
  14. J

    MX2 Integration Marathon

    Re: HSC 2018 MX2 Integration Marathon $Let $I = \int^{1}_{0}\frac{\ln(1+x)}{1+x^2}dx,$ $Put $x=\frac{1-t}{1+t}=\frac{2}{t+1}-1\;,$ Then $dx=-\frac{2}{(1+t)^2}dt $And changing limits$ $So $I = \int^{1}_{0}\frac{\ln(2)-\ln(1+t)}{1+t^2}dt=\int^{1}_{0}\frac{\ln(2)}{1+t^2}dt-I$ $So we have $2I...
  15. J

    Rational Integration

    Evaluation of \displaystyle \int\frac{x^5+3}{(x-5)^3\cdot (x-1)}dx Although i have solved it using partial fraction. But it is very lengthy so please explain me any bettter method. Thanks.
  16. J

    Limit with binomial Coefficients

    To paradoxica would you like to explain me in detail. Thanks
  17. J

    Limit with binomial Coefficients

    $Finding value of $ $\displaystyle \lim_{n\rightarrow \infty} n^{-2}\cdot \sum^{n}_{k=0}\ln\bigg(\binom{n}{k}\bigg)$
  18. J

    Trigonometric product

    $ Finding value of $ (0.5+\cos \alpha)\times(0.5+\cos 3\alpha)\times(0.5+\cos 9\alpha)\times(0.5+\cos 27\alpha)$ $ Where $\alpha =9^\circ}$ I have tried like this way $ let $(x+\cos \alpha)(x+\cos 3\alpha)(x+\cos 9\alpha)(x+\cos 27\alpha)$ $which equals $x^4+(\sum \cos \alpha)x^3+(\cos...
  19. J

    Sequence and series

    Thanks camelrider.
  20. J

    Sequence and series

    $Let $\displaystyle a_{n} = n+\frac{1}{n}$ for $n=1,2,3,4,....20$ and $p=\frac{1}{20}\sum^{20}_{n=1}a_{n}$ $And $q = \frac{1}{20}\sum^{20}_{n=1}\frac{1}{a_{n}}$. Then proving $q\in \left(0,\frac{21-p}{21}\right)$. Trial p = \frac{1}{20}\left((1+2+3+\cdot +20)+1+\frac{1}{2}+\frac{1}{3}+\cdots...
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