I got stuck on this:
Show using the binomial theorem, that if then
Hereīs my brave attempt:
So at first I thougt that each term is bigger than one (this is the part that I got wrong). Then, since there are n terms, the sum must tend to infinity because n tends to infinity and since each term bigger than one.
The b part:
I want to try this myself after getting the a) part straight but I wonder is there a clue in the b part for solving a?
Any hints, besides replacing my toilet paper with my home assignment?
Ok I think the a part is rather clear now. The only way that I see is bigger than each term though is that a sum is always bigger than any of itīs terms assuming all positive (not much proof there). Is there a more convincing way of seeing that actually is bigger than any of the terms?
To the b part. I thought that since numerator grows faster than denominator, but I quess my prof will repeat my name three times with decreasing pitch while shaking his head as he reads my conclusion.
I wanted to do a binomial expansion of the expression but canīt split the expression the same way as in part a
Maybe I donīt need to do the binomial but use what I got i part a somehow?
The assignment has a (c) part as well. Hope to get some help on that one too.
c) Show that for every
Earlier I stated that and so multiply by we get
Is this ok? And if so, how do I show that this is true for every ...induction? How?
Again it feels natural that if I keep multiplying both sides with (positive)n, k times Iīll still get left side less than right side which tends to zero as n tends to infinity.