Lim k-->∞ Σ(1+5/k)^k =?

Question

I used l'hopital's rule on this, but the derivatives just keep getting bigger and messier. I asked this question earlier today. No one solved it, but someone reminded me that the definition of e is e= lim (1+1/n)^n as n-->∞.. I don't know if this helps b/c I'm not sure how to get the 5 out from within the exponent.

Answer 1

Find Lim k→∞ of {(1+5/k)^k}

Let
h = 1/k

y = Lim k→∞ of {(1+5/k)^k}
y = Lim h→0 of {(1+5h)^(1/h)}
ln y = Lim h→0 of ln[(1+5h)^(1/h)]
ln y = Lim h→0 of {[ln(1+5h)] / h}

This is of the indeterminant form 0/0 so L'Hospital's Rule applies.

ln y = Lim h→0 of {[5/(1+5h)] / 1} = 5/(1 + 0) = 5

y = e^5

Answer 2

That sigma symbol (Σ) doesn't belong there.

Let n = k/5.

Then (1+5/k)^k = (1+1/n)^(5n) = [(1+1/n)^n]^5