Integral calculus

[charred kite]

Anyone please solve this

[dire shale]

Looks tricky. What have you tried?

And I just noticed the chapter heading that you mostly cut off.

This is why people need to include context!

[humble gazelle]

uhhhhhh

how are definite integrals involved here

Also idk how'd you do this w/o either 500 eyeQ or knowing a certain sum off the top of your head

Tbf it is a famous sum I guess

[ripe meteor]

This looks a bit arcane but I think it is not too crazy.

I think you just convert product to sum using the exponential map and then identify a riemann sum.

$$\lim_{n \to \infty} \prod_{k=1}^n \left( 1 + \frac{k^2}{n^2} \right)^{\frac{2k}{n^2}}$$
$$ = \exp \left( \lim_{n \to \infty} \frac{2}{n} \sum_{k=1}^n \frac{k}{n} \ln \left( 1 + \frac{k^2}{n^2} \right) \right)$$
$$ = \exp \left( 2 \int_0^1 dx x \ln(1 + x^2) \right)$$

[stray imp]

M8732

[ripe meteor]

[humble gazelle]

Oh ic I messed up cus I got ||e^(pi^2/3)||

My mistake was using product rule on the limit

[ripe meteor]

Did you use a different approach?

[humble gazelle]

Yeah so since every term was "going" to e^(2/m^2) as n went to infinity

Then I thought it'd be e to the sum of 1/n^2s times 2

But that's using product rule incorrectly