I FINALLY GOT THE QUESTION
It's actually easier than the last one it just looks way more cooked. Only thing is that I came across some algebraic manipulation i've never seen before but other than that it was ok.
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It's not toooo bad only comprehending the question was full cooked
after staring at this for a while, i think i finally get the part where you deal with each inequality separately
for both inequalities, we're trying to isolate e right? hence why we do the root thing to get e by itself. the confusing bit is why do we not do the same operation on the rhs? why do we raise it to the power of n+1 instead?
thats because when it comes to inequalities, as long as the inequality holds its still valid, so we dont necessarily need to do the same operation on each side
thats why they tell us that n is positive- if u raise anything >1 (which e and 1+1/n must be) to the power of n, as long as n is positive, that will be greater than if you rooted something to n.
in other words-
a^n>n√a given n>1 (if that makes sense)
this is also true for n+1
so conveniently, by raising one side to the power of n or n+1 and rooting the other side by n or n+1, we isolate e whilst getting what the question wants us to prove by getting out (1+1/n)^n or n+1 and the inequality holds
then after doing that we get e in both inequalities and we can make a sandwich inequality (as my ext 2 teacher calls them
) and boom!
