Sequences and S...

If $$n$$ is an odd integer greater than or equal to $$1$$, then the value of $$ n^3-(n-1)^3+(n-2)^3-....+(-1)^{n-1}1^3$$, is

If $$ \displaystyle f(r)=1+\frac {1}{2}+\frac {1}{3}+.....+\frac {1}{r}$$ and $$f(0)=0$$, then value of $$ \displaystyle \sum_{r=1}^{n}(2r+1)f(r)$$ is

The sum to $$n$$ terms of the series $$\displaystyle \frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+...$$ is

The sum of the first $$n$$ terms of the series $$1^2+2\cdot2^2+3^2+2\cdot4^2+5^2+2\cdot6^2+..... $$ is $$\dfrac{n \left ( n+1\right )^2}{2}$$ when $$n$$ is even. 

$$\displaystyle \frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots.n$$ terms $$=$$

Observe the following lists List I and  List II

 Sum  the  series  $$1^3+3^3+5^3+.......... $$  to  $$n$$ terms  is

ABCD is a square of length a, $$a\epsilon N$$, a>1. Let $$L_{1}$$, $$L_{2}$$, $$L_{3}$$,... be points on BC such that $$BL_{1}=L_{1}L_{2}=L_{2}L_{3}=...=1$$ and $$M_{1}$$, $$M_{2}$$, $$M_{3}$$,... be points on CD such that $$CM_{1}=M_{1}M_{2}=M_{2}M_{3}=...=1$$. Then $$\sum_{n=1}^{a-1}\left ( A{L_{n}}^{2}+L_{n}{M_{n}}^{2} \right )$$ is equal to

The sum of n terms of the series $$1 + (1 + a) + (1 + a + a^2) + (1 + a  + a^2 + a^3) + ....$$ is

If $$\displaystyle \sum_{r=1}^{n}t_{r}=\frac {n(n+1)(n+2)(n+3)}{8}$$, then $$\displaystyle \underset{n\rightarrow \infty}{ \lim} \sum_{r=1}^{n}\frac {1}{t_{r}}$$
is equal to: