HSC 2016 MX2 Integration Marathon (archive) (1 Viewer)

KingOfActing · Oct 18, 2016

Re: MX2 2016 Integration Marathon

juantheron said:
$Evaluation of $\int^{\infty}_{0}\lfloor ne^{-x} \rfloor dx,$ where $n\in \mathbb{N},n>1$

$and $\lfloor x \rfloor$ represent floor of $x$

$$We split the sum into each integer $1 \leq m \leq n $ to obtain: $\\\int_0^{\infty} \lfloor{ne^{-x}}\rfloor\,dx = \sum_{m=1}^{n-1} \int_{\ln{\left(\frac{n}{m+1} \right )}}^{\ln{\left(\frac{n}{m} \right )}}m\,dx=\sum_{m=1}^{n-1}m\ln{\left(1+\frac{1}{m} \right )}$

As far as I know, you can't simplify that sum (unless you want to turn it into a logarithm of a product).

Paradoxica · Oct 18, 2016

Re: MX2 2016 Integration Marathon

juantheron said:
$Evaluation of $\int^{\infty}_{0}\lfloor ne^{-x} \rfloor dx,$ where $n\in \mathbb{N},n>1$

$and $\lfloor x \rfloor$ represent floor of $x$

Don't post questions that don't posess closed forms.

$\sum_{k=2}^n k\log{\left( \frac{k}{k-1} \right)}$

Paradoxica · Oct 18, 2016

Re: MX2 2016 Integration Marathon

KingOfActing said:
$$We split the sum into each integer $1 \leq m \leq n $ to obtain: $\\\int_0^{\infty} \lfloor{ne^{-x}}\rfloor\,dx = \sum_{m=1}^{n-1} \int_{\ln{\left(\frac{n}{m+1} \right )}}^{\ln{\left(\frac{n}{m} \right )}}m\,dx=\sum_{m=1}^{n-1}m\ln{\left(1+\frac{1}{m} \right )}$

As far as I know, you can't simplify that sum (unless you want to turn it into a logarithm of a product).

Or the derivative of the Incomplete Zeta Function...

Drsoccerball · Oct 18, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
Don't post questions that don't posess closed forms.

$\sum_{k=2}^n k\log{\left( \frac{k}{k-1} \right)}$

Offfffttt relaxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx....

spicy_pete · Oct 30, 2016

Re: MX2 2016 Integration Marathon

Paradoxica said:
$\int_7^{13} \sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\sqrt{x+\cdots}}}}}}}}}\,$d$x$

Is it 127/6? I got that as a solution and it's right to a few decimal points (checked up to about 8 iterations of sqrt(x+sqrt(x-...)) on wolframalpha) but there are a few dubious steps

HSC 2016 MX2 Integration Marathon (archive) (1 Viewer)

KingOfActing

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