Series - Some finite series

Number of terms in the Series : n

  1. 1+2+3+...+n =\(n(n+1) \over 2\)
  2. 2+4+6+...+2n = \(n(n+1)\)
  3. 1+3+5+...+(2n-1)=\(n^2\)
  4. k+(k+1)+(k+2)+...+(k+n-1)=\(n(2k+n-1) \over 2\)
  5. \(1^2+2^2+3^2+...+n^2= {n(n+1)(2n+1) \over 6 }\)
  6. \(1^3+2^3+3^3+...+n^3= \bigg[{n(n+1) \over 2} \bigg]^2\)
  7. \(1^2+3^2+5^2+...+(2n-1)^2= {n(4n^2-1) \over 3 }\)
  8. \(1^3+3^3+5^3+...+(2n-1)^3= n^2(2n^2-1 )\)
  9. \(1+{1 \over 2}​​​​​​+​​{1 \over 4}+{1 \over 8}+...+{1 \over 2^n}+...=2\)
  10. \({1 \over 1.2 }+{1 \over 2.3 }+{1 \over 3.4 }+...+{1 \over n(n+1) }+...=1\)
  11. \(1+{1 \over 1! }+{1 \over 2! }+{1 \over 3! }+...+{1 \over (n-1)! }+...=e\)