It's magic! Putting the sum into summation notation.
Sometimes you'll be given a sum of a short list of terms, and asked to collapse the series using summation notation.
What this really means, is that you're going to assign a "term number" to each term, ie the first term, the second term, the third term, etc. Then you're going to find one function that could represent the entire sum, using a relationship between each of the original term and its position in the sum.
To do this, you'll look for patterns in the terms of the series, and bring those patterns together into the function that represents the sum. Then you'll need to find the index of the function, which is where the sum begins and ends. For example, like in this problem, the sum may begin at the first term, and end at the sixth term.
Once you've found all of this information, you'll put it together in what's called "summation notation", and the sum you write in summation notation will represent exactly the same thing as the expanded sum that you started with.