0. Sum Formulae

You should know certain sums by heart, for example:

Sum of constants

Sum of $\sum 1$

${\sum }_{i=1}^{n}1=n$

Sum of $\sum \sum 1$

${\sum }_{i=1}^n{\sum }_{i=1}^{n}1=n^2$

Sum of $\sum \sum \sum 1$

${\sum }_{i=1}^n{\sum }_{i=1}^n{\sum }_{i=1}^{n}1=n^3$

In a for loop where the inner loop depends on the outer:

Sum of $\sum \sum 1$

${\sum }_{j=1}^n{\sum }_{k=\text{j}}^{n}1={\sum }_{j=1}^n(n-j+1)=\frac{n(n+1)}{2}$

Sum of $\sum \sum 1$

${\sum }_{i=1}^n{\sum }_{k=1}^{\text{i}}1={\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}$

Sum of $i$

Sum of $i$

${\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}\le O(n^2)$

Sum of $i^2$

${\sum }_{i=1}^n{i}^2=\frac{n(n+1)(2n+1)}{6}\le O(n^3)$

Sum of $i^3$

${\sum }_{i=1}^n{i}^3=\frac{n^2(n+1{)}^2}{4}\le O(n^4)$

Logarithmic sums

Sum of $\log (i)$

${\sum }_{i=1}^n\log (i)=\log (n!)\le O(n\log (n))$

Sum of $i\log (i)$

${\sum }_{i=1}^{n}i\log (i)\le {\sum }_{i=1}^{n}n\log (n)=n^2\log n$