<s>[INST] Adding $k^2$ to $1^2 + 2^2 + \cdots + (k - 1)^2$. In his book, Calculus Vol. 1, Tom Apostol mentions that adding $k^2$ to the predicate
$$A(k): 1^2 + 2^2 + \cdots + (k - 1)^2 < \frac{k^3}{3}$$
gives the inequality
$$ 1^2 + 2^2 + \cdots + k^2 < \frac{k^3}{3} + k^2.$$
Why does the RHS $$1^2 + 2^2 + \cdots + (k - 1)^2 $$ become $$ 1^2 + 2^2 + \cdots + k^2 $$ and not $$ 1^2 + 2^2 + \cdots + (k - 1)^2 + k^2$$
when adding $k^2$? 
Thank you.
 [/INST] The notation does not matter. $1^2 + 2^2 + \cdots + (k - 1)^2 + k^2$ and $1^2 + 2^2 + \cdots + k^2$ both stand for the same value (under the appropriate interpretation of $\cdots$). Both stand for $\displaystyle \sum_{j=1}^{k}j^2$.
Example:
$1^2+2^2+3^2 + \cdots + 7^2 = 1^2+2^2+3^2 + \cdots + 6^2 + 7^2 =  1^2+2^2+3^2 + 4^2+5^2 + 6^2 + 7^2 = \displaystyle \sum_{j=1}^{7}j^2=140$
</s>