## Calculus (3rd Edition)

The sums $$\sum_{j=1}^{100} j, \quad\sum_{j=0}^{100} j$$ are equal because the extra term in the second sum is just $0$ and when you add zero to a value that value will remain unchanged. But the sums $$\sum_{j=1}^{100} 1, \quad\sum_{j=0}^{100}$$ are not equal because the extra term is equal to $1$ and this creates the difference.