 # Sum to n terms of Special Series

• The sum of first ‘n’ natural number \tt \sum n =\frac{n\left(n+1\right)}{2}
• The sum of squares of first ‘n’ natural numbers \tt \sum n^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}
• The sum of cubes of first ‘n’ natural numbers \tt \sum n^{3}=\left[\frac{n\left(n+1\right)}{2}\right]^{2}

### View the Topic in this video From 18:50 To 55:25

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Important Results and Useful Series

1.\tt \sum_{n=0}^\infty \frac{1}{n!}=e=\sum_{n=0}^\infty \frac{1}{\left(n-1\right)!}=\sum_{n=0}^\infty \frac{1}{\left(n-k\right)!}=e
2. \tt \sum_{n=1}^\infty \frac{1}{n!}=\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+....\infty = e-1
3. \tt \sum_{n=2}^\infty \frac{1}{n!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+....\infty = e-2
4. \tt \sum_{n=0}^\infty \frac{1}{\left(n+1\right)!}=\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+....\infty = e-1
5. \tt \sum_{n=1}^\infty \frac{1}{\left(n+1\right)!}=\sum_{n=0}^\infty \frac{1}{\left(n+2\right)!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+....\infty = e-2
6. \tt \sum_{n=0}^\infty \frac{1}{(2n)!}=1+=\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+....\infty = \frac{e+e^{-1}}{2}=\sum_{n=1}^\infty \frac{1}{\left(2n-2\right)!}
7. \tt \sum_{n=1}^\infty \frac{1}{\left(2n-1\right)!}=\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+.... = \frac{e-e^{-1}}{2}=\sum_{n=0}^\infty \frac{1}{\left(2n+1\right)!}
8. \tt e^{ax}=1+\frac{\left(ax\right)}{1!}+\frac{\left(ax\right)^{2}}{2!}+\frac{\left(ax\right)^{3}}{3!}+..+\frac{\left(ax\right)^{n}}{n!}+...\infty
9. \tt \sum_{n=0}^\infty \frac{n}{n!}=e=\sum_{n=0}^\infty \frac{n}{n!}
10. \tt \sum_{n=0}^\infty \frac{n^{2}}{n!}=2e=\sum_{n=1}^\infty \frac{n^{2}}{n!}
11. \tt \sum_{n=0}^\infty \frac{n^{3}}{n!}=5e=\sum_{n=1}^\infty \frac{n^{3}}{n!}
12. \tt \sum_{r=1}^n\left(a_{r}\pm b_{r}\right)=\sum_{r=1}^n a_{r} \pm \sum_{r=1}^n b_{r}
13. \tt \sum_{r=1}^n r =1+2+...+n=\frac{n\left(n+1\right)}{2}
14. \tt \sum_{r=1}^n r^{2} =1^{2}+2^{2}+3^{2}+...+n^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}
15. \tt \sum_{r=1}^n r^{3} =1^{3}+2^{3}+3^{3}+...+n^{3}=\left[\frac{n\left(n+1\right)}{2}\right]^{2}
16. \tt \sum_{r=1}^n r^{4} =1^{4}+2^{4}+3^{4}+...+n^{4}=\frac{n\left(n+1\right)n\left(6n^{3}+9n^{2}+n-1\right)}{30}
17. Sum of first n even natural numbers. i.e., 2 + 4 + 6 + .... + 2n = n (n + 1)
18. Sum of first n odd natural numbers. i.e., 1 + 3 + 5 + .... + (2n − 1) = n2
19. Sum of n terms of series 12 − 22 + 32 − 42 + 52 − 62 + 72 − 82 + .....
Case I. When n is odd = \frac{n\left(n+1\right)}{2}
Case II. When n is even = \frac{-n\left(n+1\right)}{2}