@SimplyBeautifulArt Ok thanks! I think I'll bounty a 100 rep on it to see if I can get some educated answer on the differences between the two...
In any case the student Alpha PRO is only about 60$/year so I will most likely buy it anyway since it can be accessed from the web, which is a major advantage.
I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
multiplication/division
raising to powers and roots
trigonometric functions
exponential functions
l...
Use mathematical induction to prove $n>\ln(n)$ for all integers n greater equal to $1$.
$y=\ln(x)$ is an increasing function, that is, if $a<b$, then $\ln(a)<\ln(b)$.
Using S_k and S_(k+1)
@amWhy Voted to close, I'm pretty sure I already did earlier but I can't find that flag... OP don't seem to have done any work, doesn't know ln e = 1 and keep "squeezing out" further info from answerers, that is a NO.
It isn't, I am a highschool student and I know this, we are taught in Physics how to take a derivative rather than it's definition in beginning of 11th grade. Since Limits is taught before derivatives and after calculus in Physics, we use L'Hospital's rule (remembering the standard results of derivative) directly without knowing actual definition of derivative. — Jaideep Khare29 secs ago
Anyone, just anyone tell me this is completely wrong.
hmmm... well as "anyone" I'd chime in and say that this is a bit silly but to be fair that person seems to refer to his/her personal experience with the learning of the subject...
On a side note, mirror cubes are solved with the same strategy as the 6 color 3x3 cube but it is way harder. If you really want to get crazy there are also "ghosts cubes"
Use mathematical induction to prove $n>\ln(n)$ for all integers n greater equal to $1$.
$y=\ln(x)$ is an increasing function, that is, if $a<b$, then $\ln(a)<\ln(b)$.
Using S_k and S_(k+1)
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…