MCQ
$\frac{1}{{1.2}}\,\, - \,\,\frac{1}{{2.3}}\,\, + \,\,\frac{1}{{3.4}}\,\,........\, \propto \,\,$ અનંત પદ સુધી $ = \,\,.......$
- A$2log_e2$
- B$log_22 - 1$
- C$log_e2$
- D${\log _e}\left( {\frac{4}{e}} \right)$
$S_n\,\, = \,\,\sum {{T_n}} \,\, = \,\,\sum {\frac{{{{( - 1)}^{n + 1}}}}{{n(n + 1)}}} \,\, = \,\,\sum {\frac{{{{( - 1)}^{n + 1}}}}{n}} \,\, + \,\,\sum {\frac{{{{( - 1)}^n}}}{{n + 1}}} $
$\therefore \,\,_{n \to \infty }^{\lim }\,\,{S_n}\,\, = \,\,\left( {1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \,.........\infty } \right)\,\, + \,$
$\left( { - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \,.........\infty } \right)\, - \,1$
$ = \,{\log _e}2\, + \,{\log _e}2\, - \,{\log _e}e\,\, = \,\,{\log _e}\left( {\frac{4}{e}} \right)$
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