The value of $$1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}$$ is equal to:

<p>Given,</p> <p>$$1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}}\, + \,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,....\,\, + \,11}}$$</p> <p>General term,</p> <p>${T_n} = {1 \over {1 + 2 + 3\, + \,\,....\,\, + \,\,n}}$</p> <p>$= {1 \over {{{n(n + 1)} \over 2}}}$</p> <p>$= {2 \over {n(n + 1)}}$</p> <p>$= 2\left[ {{{n + 1 - n} \over {n(n + 1)}}} \right]$</p> <p>$= 2\left[ {{1 \over n} - {1 \over {n + 1}}} \right]$</p> <p>${t_1} = 2\left[ {{1 \over 1} - {1 \over 2}} \right]$</p> <p>${t_2} = 2\left[ {{1 \over 2} - {1 \over 3}} \right]$</p> <p>${t_3} = 2\left[ {{1 \over 3} - {1 \over 4}} \right]$</p> <p>$\vdots$</p> <p>${t_n} = 2\left[ {{1 \over n} - {1 \over {n + 1}}} \right]$</p> <p>$\therefore$ ${S_n} = {t_1} + {t_2} + {t_3}\, + \,\,....\,\, + \,\,{t_n}$</p> <p>$$ = 2\left[ {{1 \over 1} - {1 \over 2} + {1 \over 2} - {1 \over 3} + {1 \over 3} - {1 \over 4}\, + \,\,...\,\, + \,\,{1 \over n} - {1 \over {n + 1}}} \right]$$</p> <p>$= 2\left[ {1 - {1 \over {n + 1}}} \right]$</p> <p>$= 2\left[ {{{n + 1 - 1} \over {n + 1}}} \right]$</p> <p>$= {{2n} \over {n + 1}}$</p> <p>$\therefore$ ${S_{11}} = {{2 \times 11} \over {11 + 1}} = {{22} \over {12}} = {{11} \over 6}$</p>