6 10 Miscellaneous Series – Class 11TH Math

6.10 Miscellaneous Series

Sigma notation: $\displaystyle\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n$ .

\sum_{k=1}^{n} k,\quad \sum_{k=1}^{n} k^2,\quad \sum_{k=1}^{n} k^3

Telescoping identity: $\displaystyle\sum_{k=1}^{n}\bigl[k^m - (k-1)^m\bigr] = n^m$ .

Properties:

\sum (a_k + b_k) = \sum a_k + \sum b_k, \qquad \sum \alpha a_k = \alpha \sum a_k

Standard formulae:

\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}, \quad \sum_{k=1}^{n} k^2 = \dfrac{n(n+1)(2n+1)}{6}, \quad \sum_{k=1}^{n} k^3 = \left[\dfrac{n(n+1)}{2}\right]^2

Sum of $1^3 + 3^3 + 5^3 + \cdots$ to $n$ terms: $T_k = (2k-1)^3$ , and

S_n = n^2(2n^2 - 1)

If $T_n = n^3 + \dfrac{3}{2}n^2 + \dfrac{1}{2}n + 1$ , then

S_n = \dfrac{n}{4}(n^3 + 4n^2 + 5n + 6)