Special Series & Sums

At a glance 12 formulas

1. 1First n numbers$1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$
2. 2Sum of squares$1^2 + 2^2 + \dots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$
3. 3Sum of cubes$1^3 + \dots + n^3 = \left[\dfrac{n(n + 1)}{2}\right]^2$$= (1 + 2 + \dots + n)^2$
4. 4Odd and even numbers$1 + 3 + \dots + (2n - 1) = n^2$$2 + 4 + \dots + 2n = n(n + 1)$
5. 5Sum not from 1$T_k + \dots + T_n = S_n - S_{k-1}$$S_n = \text{sum of the first } n \text{ terms}$
6. 6Split the general term$T_k = k^2 + k \Rightarrow \textstyle\sum T_k = \sum k^2 + \sum k$$1 \cdot 2 + 2 \cdot 3 + \dots + n(n + 1) = \tfrac{n(n+1)(n+2)}{3}$
7. 7Gaps that grow evenlyIf the gaps between terms grow by a fixed $c$, then $T_n = \frac{c}{2}n^2 + pn + q$; find $p$ and $q$ from $T_1$ and $T_2$, then sum with the standard sums.
8. 8Products of pairs$\textstyle\sum_{i<j} x_i x_j = \dfrac{(\sum x)^2 - \sum x^2}{2}$$\text{each pair of different } x\text{'s counted once}$
9. 9Telescoping sum$\dfrac{1}{k(k + d)} = \dfrac{1}{d}\left(\dfrac{1}{k} - \dfrac{1}{k + d}\right)$$\text{only the first and last pieces survive}$
10. 10Telescoping productFactor each term into neighbouring brackets, write out the first two and the last two terms, and keep only what is left after the cancelling.
11. 11Odd terms from allIf the even-$k$ terms are a fixed multiple of the whole series, the odd-$k$ terms are the rest: $\sum \frac{1}{(2k)^2} = \frac{1}{4} \sum \frac{1}{k^2}$.
12. 12AGP to infinity$S_\infty = \dfrac{a}{1 - r} + \dfrac{dr}{(1 - r)^2}$$a,\ (a + d)r,\ (a + 2d)r^2, \dots,\ |r| < 1$