Fun-time-1-2x-3x-2-4x-3-1-1-x-2-1-4-12-32-1-1-2-2-4-12-32-0-No-1-fun-5-11-17-23-0-n-1-6n-1-6-n-1-n-1-6-1-12-1-2-0-n-1-12-

Question Number 107101 by Dwaipayan Shikari last updated on 08/Aug/20

Fun time

1 + 2 x + {3 x}^{2} + {4 x}^{3} + \dots . = \frac{1}{{(1 - x)}^{2}}

1 + 4 + 12 + 32 + \dots = \frac{1}{{(1 - 2)}^{2}}

4 + 12 + 32 + \dots . = 0 (No 1 fun)

5 + 11 + 17 + 23 + \dots = 0

\underset{n = 1}{\sum^{\infty}} 6 n - 1 = 6 \underset{n = 1}{\sum^{\infty}} n - \sum^{\infty} 1 = 6 . (- \frac{1}{12}) - (- \frac{1}{2}) = 0

\sum^{\infty} n = - \frac{1}{12} (Ramanujan sum)

\sum^{\infty} 1 = 1 + 1 + 1 + 1 + 1 + \dots = - \frac{1}{2}

\sum^{\infty} n^{2} . \sum^{\infty} \frac{1}{n^{2}} ⩾ {(\sum^{\infty} 1)}^{2} (Cauchy schwarz ineqality)

\sum^{\infty} n^{2} . \frac{π^{2}}{6} ⩾ \frac{1}{4}

\sum^{\infty} n^{2} ⩾ \frac{3}{2 π^{2}}