Dumb question: what is $\sum\limits_{n = 1}^k 1$ Let $\{I_n\}$ be a collection of intervals on $\Bbb {R}$, whose length denoted by $I_n = |I_n|$
Then what is $\sum\limits_{n = 1}^k (I_n + \alpha)$, where $\alpha$ is some real number?
The notation confuses me:
$$\sum_{n = 1}^k (I_n + \alpha) = \sum_{n = 1}^k I_n +  \sum_{n = 1}^k \alpha = \sum_{n = 1}^k I_n +  \alpha \sum_{n = 1}^k 1$$
What is $\sum\limits_{n = 1}^k 1$?
Confusion 1: $n$ was an index, there is no index in the $\sum\limits_{n = 1}^k 1$ term...
Confusion 2:$\sum\limits_{n = 1}^k 1 = 1 + 1 +\cdots + 1 = k$? or does that sum term does nothing since it is not summing over something indexed by $n$?$

You can't "sum intervals". I believe that $I_k+\alpha$ means, if $I_k=(a_k,b_k)$, the interval $(a_k+\alpha, b_k+\alpha)$
