Binomial Expansion as n gets very large

1 + r + (1-\dfrac{1}{n})\dfrac{r^2}{2} +(1-\dfrac{1}{n})\dfrac{r^3}{3!} + ... + (1-\dfrac{1}{n})\dfrac{1}{k!}r^k

Are you sure it doesn't say:

$<1 + r + (1-\dfrac{1}{n})\dfrac{r^2}{2} +(1-\dfrac{1}{n})\dfrac{r^3}{3!} + ... + (1-\dfrac{1}{n})\dfrac{1}{k!}r^k$

Note the initial inequality.

No, there is an equals sign.

No, there is an equals sign.

Then you have a typo. Plug in some small numbers and evaluate the expression, and you'll see they're not equal.

Are you sure it doesn't say:

$<1 + r + (1-\dfrac{1}{n})\dfrac{r^2}{2} +(1-\dfrac{1}{n})\dfrac{r^3}{3!} + ... + (1-\dfrac{1}{n})\dfrac{1}{k!}r^k$

Note the initial inequality.

I don't think the issue can be inequality v.s. equality, because (I assume) the aim is to show that the limit is

$1 + r + r^2/2! + r^3/3! + ...$

This is true, we are aiming to show the limit is the above

This is true, we are aiming to show the limit is the above

In my Analysis lecture, we were looking at the limit as n approaches infinity of

$(1 + \dfrac{r}{n})^n$

So by applying the binomial formula we get

$1 + n\dfrac{r}{n} + \dfrac{n(n-1)r^2}{2n^2} + \dfrac{n(n-1)(n-2)r^3}{6n^3} +... +\dfrac{n(n-1)...(n-k+1)}{k!}\dfrac{r^k}{n^k}[br]$

I am happy with that, this is where I'm not happy. I agree with the first three terms of this sum but no more, so to clarify it's r^3 term and beyond I am unhappy with.

$1 + r + (1-\dfrac{1}{n})\dfrac{r^2}{2} +(1-\dfrac{1}{n})\dfrac{r^3}{3!} + ... + (1-\dfrac{1}{n})\dfrac{1}{k!}r^k$

I can see that $\dfrac{n(n-1)(n-2)}{n^3}$ tends to 1, as n approaches infinity, and I have seen graphically it approaches $1-\dfrac{1}{n}$. But I cannot see how it is done algebraically. Also, I'm not clear on how the general term becomes $(1-\dfrac{1}{n})\dfrac{1}{k!}r^k$

Thanks for your replies.

I went to a similar or the same lecture. In my notes I have it as
$1 + r + (1-\dfrac{1}{n})\dfrac{r^2}{2} +(1-\dfrac{1}{n})(1-\dfrac{2}{n})\dfrac{r^3}{3!} + ... + (1-\dfrac{1}{n})(1 - \dfrac{2}{n})...(1-\dfrac{k-1}{n})\dfrac{1}{k!}r^k +...$



Is by any chance the second lecture now?

I am doing M137 @ Warwick.

Yes, this is a correct equality. Seems a bit of a drastic typo (in the OP./ lecture) to omit about half the bracketed terms ?!?

As am I.



I think there was a bit of a rush to get everything down in the last few seconds of the lecture(all these long sums) and that step and the next weren't well explained so OP may have made some mistakes copying it down. Also, I can't remember what the lecturer wrote on the board, so if there was a typo on the board I might not have written it down exactly, but written the closest thing that made sense.
I think it was just to give an example of a limit before making the course becomes more formal.