I will just show a classic usage of theorem of Laplace here. Let f(x)=ln(x) f: R+ -> R . As an elementary function it is continuous and derivable on (0,+inf) and subsequently on each interval [k,k+1] where k is a natural number. Since the prerequisites are satisfied from applying the Theorem of Laplace it follows that for each interval [k,k+1] there exists a number c such that k
Pretty much what the Theorem of Laplace usage does. You take a term on the series and integrate it or just know what it is the derivative of and go through the proof. You can use it to prove 1/√n diverges or series like n^(-r) when r
I prefer the standard proof:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ...
> 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ...
= 1 + 1/2 + 1/2 + 1/2 + ...
which obviously diverges.
Yesh, I like this one more too! It just wasn’t the first thing that sprang to my mind at the time.
@@mathoutloud Hi, thanks for the doing question, the solution given above was the given answer.
this is so real analysis i love it. i literally just took the class and this was my first thought when i saw this problem just now haha
I will just show a classic usage of theorem of Laplace here.
Let f(x)=ln(x) f: R+ -> R . As an elementary function it is continuous and derivable on (0,+inf) and subsequently on each interval [k,k+1] where k is a natural number. Since the prerequisites are satisfied from applying the Theorem of Laplace it follows that for each interval [k,k+1] there exists a number c such that k
I like the method of comparing to the series with powers of 2, it feels a little overkill to resort to integration like I did.
Pretty much what the Theorem of Laplace usage does. You take a term on the series and integrate it or just know what it is the derivative of and go through the proof.
You can use it to prove 1/√n diverges or series like n^(-r) when r