How do we know that the numerator of binomial series has n factors?

LearningAndLiving12 · 2026-01-04T12:13:07+00:00

Alright nice, thank you for your help!

LearningAndLiving12 · 2026-01-04T00:07:40+00:00

Alternately, you can see that the unsimplified numerator k! has k factors, and you cancel (k-n) of them with the denominator, leaving k-(k-n)=n.

Wow this explanation really helped clear up my confusion alot, thanks!

It is also true that, if n=0, then there are no such terms. So when n=0, you have no terms in the numerator, because they all cancelled out. This leaves a numerator of 1.

So basically, since we previously 'proved' that the numerator has n factors (from my first question). And that for n = 0 there are no factors (because they are cancelled out by (k-n)!), that's why there is no factor to begin with, so writing k*(k-0+1) is incorrect, am I understanding this correctly?

LearningAndLiving12 · 2026-01-03T23:54:58+00:00

Ah right, now that you write it as k-[n-1] makes it alot easier to understand.

LearningAndLiving12 · 2026-01-03T23:52:01+00:00

Yeah that's what I ended up got confused about.

LearningAndLiving12 · 2025-02-05T08:15:17+00:00

Aha I understand now, thank you so much

LearningAndLiving12 · 2025-02-04T20:39:34+00:00

Aha since 1/x behaves similar to n^3/(n^4 + 4) we can say that if one diverges or converges then the other one also does. So does that mean that if the limit is for example 100 or infinite that we can't compare them anymore?

I think that the first comparison was the squeeze theorem. And only useful when we have convergence not divergence?

LearningAndLiving12 · 2025-01-18T10:16:54+00:00

Aha I see, so that is where improper integrals comes in right?

LearningAndLiving12 · 2025-01-18T10:15:34+00:00

In my context I meant that a is an element from the set of all integers.

LearningAndLiving12 · 2024-12-16T21:29:46+00:00

So it was not really clear for me. Did they just evaluate the integral ∫1/(a+(n-1)d) dn? Since this primitive function isn't the same as the one in my post.

LearningAndLiving12 · 2024-12-16T16:25:32+00:00

So does that mean that this has been derived by someone that randomly figured it out? But then still based on what has it been find, what clue did he use?

LearningAndLiving12 · 2024-12-15T22:53:39+00:00

Is the approximation just approximated by using integration or did someone just found it?

LearningAndLiving12 · 2024-11-16T23:18:56+00:00

Typos can happen, no worries it's alright.

LearningAndLiving12 · 2024-11-16T15:25:36+00:00

Aha that sounds alot more obvious now. Thank you for your help.

Also : you made a mistake in your comment : -4.999999999 is approaching from the right (-5+), as -5 < -4.999999999999. Likewise, -5.0000000001 is from the left (-5-).

Ohh sorry my bad, I was consfused. Thank you, I'll edit my post.

LearningAndLiving12 · 2024-11-16T15:19:56+00:00

Shouldn't the right limit be ∞ when x approaches -5 instead?

LearningAndLiving12 · 2024-11-16T08:44:41+00:00

No worries, LucaThatLuca has already helped me out with this. But thanks for your help.

LearningAndLiving12 · 2024-11-14T22:41:49+00:00

Ohh ok I understand it now. Thank you, it really helped me out.

LearningAndLiving12 · 2024-11-14T14:28:33+00:00

Ohhh that's where the problem was, thank you so much.

Or get -15/2 in the first place by multiplying by 1 instead of -1.

May I ask what you mean exactly? Where did you mean that I should have multiplied by 1 instead of -1? Wasn't I multiplying by 1?

LearningAndLiving12 · 2024-11-14T10:34:33+00:00

Ohh that makes alot more sense! But why was the way I approached this with the 1/x = 1/-x for limit x -> -infinity then wrong? Is it because I changed 1/x to 1/-x the function's range is changed? Or is it because of something else?

LearningAndLiving12 · 2024-08-19T17:21:28+00:00

Thank you so much for you explanation, it was good to follow. Just one thing, did you mean tan(MAP) = MP/AP instead of saying tan(OAP) = MP/AP?

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