Finding limit and convergence of integral

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[river raft]

Is it the integral of x^2n/(x^2+x+3 + cos(x)^2024) ?

If so then the the integral definitely converges because the function is continuous on [0;1]

[river raft]

Oh you need to calculate the limit for n approaching infinity ?

Here, if x is on the interval [0,1[ what’s the limit of x^2n as n approaches infinity ?

It’s fine dw

Yes but when you integrate on [0;1[ or [0,1] it’s the same because continuous by parts function when integrated on an interval are equal by finite parts differences so if you integrate on [0,1] or on [0,1[ it’s the same

Yes

But you need to justify interverting the limit and integral

ie use the dominated convergence theorem, have you heard of it ?

Well here if you want to find the limit you need to consider the limit inside the integral

[river raft]

The problem is you aren’t guaranteed to be allowed to switch limit and integral here it works but still

Haven’t heard of it what’s the claw criterion?

Oh it’s fine if you haven’t heard of it it’s my bad you must use another method then

Oh okay I see

Well here you can see that the integral is always positive

And consider that 1/(x^2+x+cos(x)^2024 +3) <=1

For any x on that interval

Yes

You get 1/(2n+1)

[spring lion]

@copper phoenix has given 1 rep to @river raft

[river raft]

Np I’m sorry I should have thinked simpler lol idk why I used complicated theorems