Sequences and S...

If $$ s_n = \displaystyle \sum_{r=0}^n \dfrac {1}{^nC_r}$$ and $$ t_n = \displaystyle \sum_{r=0}^n \dfrac {r}{^nC_r} , $$ then $$ \dfrac {t_n}{s_n} $$ is equal to - 

$$\displaystyle \sum_{r = 0}^{n}{t^3 \left( \frac{^nC_r}{^nC_{r-1}} \right)^2 }$$ is equal to

If $$x=\dfrac{1}{5}+\dfrac{1.3}{5.10}+\dfrac{1.3.5}{5.10.15}+.....\infty$$ then $$3x^2+6x=$$

The value of $$\sum _{ n=1 }^{ 9999 }{ \cfrac { 1 }{ \left( \sqrt { n } +\sqrt { n+1 }  \right) \left( \sqrt [ 4 ]{ n } +\sqrt [ 4 ]{ n+1 }  \right)  }  } $$ is

Let $${ T }_{ r }$$ and $${ S }_{ r }$$ be the $${ r }^{ th }$$ term and sum up to $${ r }^{ th }$$ term of a series respectively. If for an odd natural number $$n,{ S }_{ n }=n$$ and $${ T }_{ n }=\dfrac { { T }_{ n-1 } }{ { n }^{ 2 } }$$, then $${ T }_{ m }$$ ($$m$$ being even) is:

Sum of the series $$\displaystyle\sum^n_{r=1}(r^2+1)r!$$ is?

In a certain code language, DIPLOMA is written as FERHQIC, then what is the code for PENCILS in the language? 

If $$x\in R$$ and $$S=1-{ C }_{ 1 }\cfrac { 1+x }{ { \left( 1+nx \right)  }^{  } } +{ C }_{ 2 }\cfrac { 1+2x }{ { \left( 1+nx \right)  }^{ 2 } } -{ C }_{ 3 }\cfrac { 1+3x }{ { \left( 1+nx \right)  }^{ 3 } } +...upto\quad (n+1)$$ terms, then $$S$$

The sum to infinite of the series
$$S=1+\cfrac { 2 }{ 3 } +\cfrac { 6 }{ { 3 }^{ 2 } } +\cfrac { 6 }{ { 3 }^{ 3 } } +\cfrac { 6 }{ { 3 }^{ 4 } } +.....\quad $$ is