Bizarre Limit Question

[Phovesmos]

I was investigating a claim made in a textbook surrounding an implication that follows from the definition of e.

In particular, I was trying to prove the following:



Let lambda = p

I tried letting y = (1+(1/n))^(-pn), but this has proven, thus far, to be a dead end.

I think the path might be to mess around with binomial expansions to expand the right hand limit to eventually equal the Taylor expansion of e^-p... But this is proving rather nasty... So... Here's what I have so far:

Attempt #1 (Used y-Substitution and messed around--FAILED):





Attempt #2 (Equating binomial expansion to Taylor Expansion):


This turned into something ridiculous and only served to magnify the quirks of infinite series. I tried to recover by evaluating from an earlier point (Recovery A), but even though we can get close to the answer, we're stuck with a coefficient that cannot, in any way, be equated to the Taylor Expansion of e^-p. Observe:

This post continues in a self-response. I am only able to upload four images.

 

[Phovesmos]

Expanding to show the "other side" of the limit (which is a bit ludicrous leads to a dead end. nevertheless... here I try:



We're still stuck with the nasty coefficient. Finally, I've tried to equate the binomial expansion to the Taylor Expansion in hopes that maybe something will arise:



Please help me. I'm royally stuck.

 

[pka]

This is a standard result in almost any analysis text.
\((a > 0,b > 0,\left( {\forall n} \right)\left[ {nb + c \ne 0} \right])\)\(\mathop {\lim }\limits_{n \to \infty } {\left( {1 \pm \dfrac{a}{{bn + c}}} \right)^n} = {e^{\pm\frac{{ a}}{b}}}\)
The proof is based upon the fact that \(\mathop {\lim }\limits_{n \to \infty } {\left( {1 + \dfrac{t}{n}} \right)^n} = {e^t}\).

 

[Phovesmos]

I found the following to answer my question:

https://math.stackexchange.com/questions/882741/limit-of-1-x-nn-when-n-tends-to-infinity

BUT!

There's a massive problem in the work that I've done... Does my work imply that the binomial theorem cannot be applied to a binomial when the limit of a binomial's exponent approaches infinity? That's the only conclusion that I can draw based on the math that I've done. If the binomial theorem did apply, then the coefficients of the binomial expansion should be equal to the coefficients of the Taylor Expansion, but they're not. Because of this, there exists a contradiction, which implies the inability to apply the binomial theorem to a binomial with an exponent whose limit approaches infinity.

But still... There's something off-putting about this. My math clearly shows the ability to construct a coefficient that varies from what we see. Do you have any insight into what's going on here? I've spent hours messing around with this and it would truly relieve a burden if there was even a slight push in the right direction that could shed light into what's going on.

 

[Dr.Peterson]

I was investigating a claim made in a textbook surrounding an implication that follows from the definition of e.

In particular, I was trying to prove the following:

View attachment 23605

Let lambda = p

I tried letting y = (1+(1/n))^(-pn), but this has proven, thus far, to be a dead end.

Try a substitution: [MATH]e^t=\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\right]^t[/MATH] and let [MATH]m=nt[/MATH] ...