is it meant to be sin54cos26?Not sure if this question works but oh well
Not sure if this question works but oh well
Tried to fix for you.Having a stab at that one...It's probably fudged...but just trying.
A classic fallacious argument in analysis. The number of terms in this series is not fixed, so we cannot simply take the limits of each term individually and add them. If things like this were legit then we would have for exampleDon't worry carrot lol. For some reason lately my upload speed has been terrible so I can't even upload a simple gif i downloaded off codecogs.
ANYWAY
What I did was Consider (1+1/n)^n and expand with binomial theorem: Cn0 +Cn1(1/n)+Cn2(1/n^2)=...
Then expand out into factorial notation
= 1+ n!/1!(n-1)! * 1/n ...
=1+ 1/1! + (n-1)/2!*n +...
=1+1/1! + (1/2!)* (1+1/n) and the rest of the brackets end up with 1 +/- x/n, n^2 so they all disappear once we take the limit as n=> infinity since 1/n goes off to 0. So you're left with 1+1/1!+1/2!+1/3!... which is the required result.
A little iffy about that question, because the NSW Syllabus only ever defines integral values of n (for the Binomial Expansion)...
Have fun!
That is all this question needs , but it is sort of uni analysis-y.A little iffy about that question, because the NSW Syllabus only ever defines integral values of n (for the Binomial Expansion)...
This is the problem with your argument Sy123, you cannot treat such a series "term by term". There isn't really a proof of this equality that a 3U student could be expected to produce...I will post a sandwich argument a bit later.A classic fallacious argument in analysis. The number of terms in this series is not fixed, so we cannot simply take the limits of each term individually and add them. If things like this were legit then we would have for example
lim (1+1/n)^n = lim 1^n = 1, as each term in the n-fold product (1+1/n)(1+1/n)...(1+1/n) tends to 1.