2015-16 the most fruitful years of my life!

Calculus Level 5

If, $A=\int _{ 0 }^{ 1 }{ \left( \sum _{ i=0 }^{ 2014 }{ { x }^{ i } } \right) dx } -\int _{ 0 }^{ \infty }{ \left( \frac { { e }^{ -t }-{ e }^{ -2t }-t{ e }^{ -2016 } }{ t-t{ e }^{ -t } } \right) dt }$

Find: $\\ \left\lfloor 10000A \right\rfloor$

Try more here!

The answer is 5772.

1 solution

Tanishq Varshney
Sep 30, 2015

$\large{\psi \left( s+1 \right) =-\gamma +\displaystyle\int _{ 0 }^{ 1 }{ \frac { 1-{ x }^{ s } }{ 1-x } } dx\\ \\ \psi \left( s \right) =\displaystyle \int _{ 0 }^{ \infty }{ \left( \frac { { e }^{ -t } }{ t } -\frac { { e }^{ -st } }{ 1-{ e }^{ -t } } \right) } dt}$

where $\gamma$ is $\text{Euler Mascheroni constant}$ and its value is equal to $0.5772156$ .

$\large{A=\gamma}$

Make it interesting a little bit. Prove both the statements that you have written. And I have something different too. Try to find the answer without even considering digamma function since the answer doesn't include it.

Kartik Sharma - 5 years, 8 months ago

Solution without digamma function will become too tedious.