The value of $\overline{0.037}$,where $\overline{0.037}$ stands for the number $0.037037037...$,is

The sum to $20$ terms of the series $2^2-3^2+4^2-5^2+6^2-\ldots$ is equal to $........$.

If $0 < \theta, \phi < \frac{\pi}{2}$,$x = \sum_{n=0}^{\infty} \cos^{2n} \theta$,$y = \sum_{n=0}^{\infty} \sin^{2n} \phi$,and $z = \sum_{n=0}^{\infty} \cos^{2n} \theta \cdot \sin^{2n} \phi$,then:

The ${n^{th}}$ term of the series $\frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \dots$ is:

The greatest integer less than or equal to the sum of the first $100$ terms of the sequence $\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots$ is equal to

If $\sum_{k=1}^n \left( \sum_{m=1}^k m^2 \right) = an^4 + bn^3 + cn^2 + dn + e$,then which of the following is true?