Can someone help me understand this infinite product question?

Question

Can someone walk this through for me, so I understand how to get from $A$ to $B$?

$$\prod_{k=1}^K e^k=e^{K(K+1)/2}$$

Answer 1

To obtain the solution, as indicated in the comments, we need to use the following results

• $e^x\cdot e^y=e^{x+y}$

• $\sum_{k=1}^K k=\frac{K(K+1)}2$

Answer 2

The product of bases to powers is the base to the sum of the powers.

Or in other words $b^n b^m = b^{n+m}$.

That's all.

......

So $e^1e^2e^3.......e^K = e^{1+2+3+...... K}$.

That's all.

.......

So $\prod_{k=1}^K e^k = e^{\sum_{k=1}^K k}$.

That's all.

.....

And $\sum_{k=1}^k k = \frac {K(K+1)}2$ is a well known result everyone is expected to know.

Answer 3

$$a_K=\prod_{k=1}^K e^k \implies \log(a_K)=\sum_{k=1}^K\log(e^k)=\sum_{k=1}^Kk=\frac{K(K+1)}2\implies a_K=e^{K(K+1)/2}$$