I have a basic question about sums which I can't seem to get intuition for their start and end points. $$ \sqrt[n]{e} \sum_{k=0}^{n-1} e^{k/n} = \sum_{k=1}^n e^{k/n}\\ \sqrt[n]{e} \sum_{k=1}^{n-1} e^{k/n} = \sum_{k=1}^n e^{k/n} $$
So the first equation is true, the second isn't. By my intuition the second would be true and the first false, but it isn't so.
How I understand it: (Look at the right side of the second equation)The "endpoint" of the sum is $n$ and the start point is $k=1$. If we factor out the $e^{\frac{1}{n}}$. Wouldn't the "startpoint" stay the same so $k=1$ and the endpoint would shift for one "downwards", so we get the "endpoint" $n-1$
Wouldn't it be so if we move everything by one "downwards" we get "startpooint" $k=0$ and "endpoint" $n=n-1$ the sums would stay the same , and we wouldn't count in the factored out $e^{\frac{1}{n}}$, which in this case should be .
So can someone explain why my intuition is false and the first equation is true.
When you talk about shifting the start point and end point, you'll either shift both of them or neither of them. It won't make sense if you shift one but not the other. I'll write it out in detail for you. We have $$\sum_{k=1}^n e^{k/n}=e^{1/n}\sum_{k=1}^n e^{(k-1)/n}$$ just by pulling out a factor of $e^{1/n}$. Now we want to adjust the indices. To make it plainer I'll use a new index, say $j=k-1$. Now when $k=1, j=0$ and when $k=n, j=n-1$ so we have $$ e^{1/n}\sum_{k=1}^n e^{(k-1)/n}=e^{1/n}\sum_{j=0}^{n-1} e^{j/n}$$