Clarification: Summation step

Question

I am reading Concrete Mathematics by Donald Knuth. There is a method to solve summation, particularly for $\sum_{k=0}^n k^2$, called Expand and contract.

I don't understand just a single step. i.e. $$\sum_{1\leq k \leq n} k^2 = \sum_{1\leq j \leq k \leq n} k$$

score 3 · Accepted Answer · answered Aug 07 '17 at 17:52

3

The idea is to say that $k = \sum_{j=1}^k 1$

$$\sum_{1\leq k \leq n} k^2 = \sum_{1\leq k \leq n} k \left(\sum_{j=1}^k 1\right)= \sum_{1\leq j \leq k \leq n} k$$

answered Aug 07 '17 at 17:52

fonfonx

4,282

I don't understand you're saying $k = \frac {k(k + 1)}{2}$! – Abhisek Aug 07 '17 at 17:55
@Abhisek where? – fonfonx Aug 07 '17 at 17:57
1

my bad, I didn't read it properly – Abhisek Aug 07 '17 at 17:59

Clarification: Summation step

1 Answers1