JEE Main & Advanced Mathematics Sequence & Series Question Bank

done Logarithmic series Total Questions - 36

• question_answer1)
  \[\frac{2}{1}\,.\,\frac{1}{3}+\frac{3}{2}.\frac{1}{9}+\frac{4}{3}.\frac{1}{27}+\frac{5}{4}.\frac{1}{81}+......\infty =\]
  A)
  \[\frac{1}{2}-{{\log }_{e}}\frac{2}{3}\] done clear
  B)
  \[-{{\log }_{e}}\frac{2}{3}\] done clear
  C)
  \[\frac{1}{2}+{{\log }_{e}}\left( \frac{2}{3} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[\frac{1}{2}{{x}^{2}}+\frac{2}{3}{{x}^{3}}+\frac{3}{4}{{x}^{4}}+......\infty =\]
  A)
  \[\frac{x}{1+x}-{{\log }_{e}}(1-x)\] done clear
  B)
  \[\frac{x}{1+x}+{{\log }_{e}}(1-x)\] done clear
  C)
  \[\frac{x}{1-x}-{{\log }_{e}}(1-x)\] done clear
  D)
  \[\frac{x}{1-x}+{{\log }_{e}}(1-x)\] done clear
  View Solution play_arrow

• question_answer3)
  \[\frac{x-1}{(x+1)}+\frac{1}{2}\,.\,\frac{{{x}^{2}}-1}{{{(x+1)}^{2}}}+\frac{1}{3}\,.\,\frac{{{x}^{3}}-1}{{{(x+1)}^{3}}}+......\infty =\]
  A)
  \[{{\log }_{e}}x\] done clear
  B)
  \[{{\log }_{e}}(1+x)\] done clear
  C)
  \[{{\log }_{e}}(1-x)\] done clear
  D)
  \[{{\log }_{e}}\frac{x}{1+x}\] done clear
  View Solution play_arrow

• question_answer4)
  \[\frac{1}{1\,.\,2\,.\,3}+\frac{1}{3\,.\,4.\,5}+\frac{1}{5\,.\,6.\,7}+.....\infty =\]
  A)
  \[{{\log }_{e}}\sqrt{2}\] done clear
  B)
  \[{{\log }_{e}}2-\frac{1}{2}\] done clear
  C)
  \[{{\log }_{e}}2\] done clear
  D)
  \[{{\log }_{e}}4\] done clear
  View Solution play_arrow

• question_answer5)
  \[\frac{1}{2}+\frac{3}{2}\,.\,\frac{1}{4}+\frac{5}{3}.\frac{1}{8}+\frac{7}{4}.\frac{1}{16}+.....\infty =\]
  A)
  \[2-{{\log }_{e}}2\] done clear
  B)
  \[2+{{\log }_{e}}2\] done clear
  C)
  \[{{\log }_{e}}4\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  \[{{\log }_{e}}\sqrt{\frac{1+x}{1-x}}=\]
  A)
  \[{{\log }_{e}}\frac{1}{2}\] done clear
  B)
  \[2\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+.....\infty  \right]\] done clear
  C)
  \[2\,\left[ {{x}^{2}}+\frac{{{x}^{4}}}{4}+\frac{{{x}^{6}}}{6}+.....\infty  \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  \[\frac{1}{x+1}+\frac{1}{2\,{{(x+1)}^{2}}}+\frac{1}{3\,{{(x+1)}^{3}}}+....\infty =\]
  A)
   \[{{\log }_{e}}\left( 1+\frac{1}{x} \right)\] done clear
  B)
  \[{{\log }_{e}}\left( 1-\frac{1}{x} \right)\] done clear
  C)
  \[{{\log }_{e}}\left( \frac{x}{x+1} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  \[{{\log }_{e}}(x+1)-{{\log }_{e}}(x-1)=\]
  A)
  \[2\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+......\infty  \right]\] done clear
  B)
   \[\,\left[ x+\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}+......\infty  \right]\] done clear
  C)
  \[2\,\left[ \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+...\infty  \right]\] done clear
  D)
   \[\,\left[ \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+...\infty  \right]\] done clear
  View Solution play_arrow

• question_answer9)
  \[\left( \frac{a-b}{a} \right)+\frac{1}{2}{{\left( \frac{a-b}{a} \right)}^{2}}+\frac{1}{3}\,{{\left( \frac{a-b}{a} \right)}^{3}}+.....=\] [MNR 1979; MP PET 1990; UPSEAT 2001, 02; AMU 2005]
  A)
  \[{{\log }_{e}}(a-b)\] done clear
  B)
  \[{{\log }_{e}}\left( \frac{a}{b} \right)\] done clear
  C)
  \[{{\log }_{e}}\left( \frac{b}{a} \right)\] done clear
  D)
   \[{{e}^{\left( \frac{a-b}{a} \right)}}\] done clear
  View Solution play_arrow

• question_answer10)
  \[\frac{(a-1)-\frac{{{(a-1)}^{2}}}{2}+\frac{{{(a-1)}^{3}}}{3}-....\infty }{(b-1)-\frac{{{(b-1)}^{2}}}{2}+\frac{{{(b-1)}^{3}}}{3}-.....\infty }=\]
  A)
  \[{{\log }_{b}}a\] done clear
  B)
  \[{{\log }_{a}}b\] done clear
  C)
  \[{{\log }_{e}}a-{{\log }_{e}}b\] done clear
  D)
  \[{{\log }_{e}}a+{{\log }_{e}}b\] done clear
  View Solution play_arrow

• question_answer11)
  \[\frac{1}{5}+\frac{1}{2}\,.\,\frac{1}{{{5}^{2}}}+\frac{1}{3}.\frac{1}{{{5}^{3}}}+.....\infty =\]
  A)
  \[{{\log }_{e}}\frac{4}{5}\] done clear
  B)
  \[{{\log }_{e}}\frac{\sqrt{5}}{2}\] done clear
  C)
  \[2{{\log }_{e}}\frac{\sqrt{5}}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  \[{{\log }_{e}}\,[{{(1+x)}^{1+x}}{{(1-x)}^{1-x}}]\,=\]
  A)
  \[\frac{{{x}^{2}}}{2}+\frac{{{x}^{4}}}{4}+\frac{{{x}^{6}}}{6}+....\infty \] done clear
  B)
  \[\frac{{{x}^{2}}}{1.2}+\frac{{{x}^{4}}}{3.4}+\frac{{{x}^{6}}}{5.6}+....\infty \] done clear
  C)
  \[2\,\,\left[ \frac{{{x}^{2}}}{1.2}+\frac{{{x}^{4}}}{3.4}+\frac{{{x}^{6}}}{5.6}+..\infty  \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  In  the expansion of  \[2{{\log }_{e}}x-{{\log }_{e}}(x+1)-{{\log }_{e}}(x-1)\], the coefficient of \[{{x}^{-4}}\] is
  A)
  1/2 done clear
  B)
  \[-1\] done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  The sum of the series\[\frac{1}{2\,.\,3}+\frac{1}{4\,.\,5}+\frac{1}{6\,.\,7}+...=\] [MP PET 1998]
  A)
  \[\log \,(2/e)\] done clear
  B)
  \[\log \,(e/2)\] done clear
  C)
  2/e done clear
  D)
  e/2 done clear
  View Solution play_arrow

• question_answer15)
  If \[b=a-\frac{{{a}^{2}}}{2}+\frac{{{a}^{3}}}{3}-\frac{{{a}^{4}}}{4}+..\]then \[b+\frac{{{b}^{2}}}{2\,!}+\frac{{{b}^{3}}}{3\,!}+\frac{{{b}^{4}}}{4\,!}+...\infty =\]
  A)
  \[{{\log }_{e}}a\] done clear
  B)
  \[{{\log }_{e}}b\] done clear
  C)
  \[a\] done clear
  D)
  \[{{e}^{a}}\] done clear
  View Solution play_arrow

• question_answer16)
  \[\frac{1}{1\,.\,2}-\frac{1}{2\,.\,3}+\frac{1}{3\,.\,4}-\frac{1}{4\,.\,5}+.....\infty =\] [Roorkee 1992; AIEEE 2003]
  A)
  \[{{\log }_{e}}\frac{4}{e}\] done clear
  B)
  \[{{\log }_{e}}\frac{e}{4}\] done clear
  C)
  \[{{\log }_{e}}4\] done clear
  D)
  \[{{\log }_{e}}2\] done clear
  View Solution play_arrow

• question_answer17)
  \[1+\left( \frac{1}{2}+\frac{1}{3} \right)\,\frac{1}{4}+\left( \frac{1}{4}+\frac{1}{5} \right)\,\frac{1}{{{4}^{2}}}+\left( \frac{1}{6}+\frac{1}{7} \right)\,\frac{1}{{{4}^{3}}}+....\infty =\]
  A)
  \[{{\log }_{e}}(2\sqrt{3})\] done clear
  B)
  \[2\,\,{{\log }_{e}}2\] done clear
  C)
  \[{{\log }_{e}}2\] done clear
  D)
  \[{{\log }_{e}}\left( \frac{2}{\sqrt{3}} \right)\] done clear
  View Solution play_arrow

• question_answer18)
  \[\frac{1}{1\,.\,3}+\frac{1}{2}\,.\,\frac{1}{3\,.\,5}+\frac{1}{3}\,.\,\frac{1}{5\,.\,7}+......\infty =\]
  A)
  \[2\,{{\log }_{e}}2-1\] done clear
  B)
  \[{{\log }_{e}}2-1\] done clear
  C)
  \[{{\log }_{e}}2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  \[\frac{4}{1\,.\,3}-\frac{6}{2.4}+\frac{12}{5\,.\,7}-\frac{14}{6\,.\,8}+.....\infty =\]
  A)
  \[{{\log }_{e}}3\] done clear
  B)
  \[{{\log }_{e}}2\] done clear
  C)
  \[2\,{{\log }_{e}}2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  \[{{\log }_{e}}x-{{\log }_{e}}(x-1)=\]
  A)
  \[\frac{1}{x}-\frac{1}{2{{x}^{2}}}+\frac{1}{3{{x}^{3}}}-.....\infty \] done clear
  B)
  \[\frac{1}{x}+\frac{1}{2{{x}^{2}}}+\frac{1}{3{{x}^{3}}}+.....\infty \] done clear
  C)
  \[2\,\left( \frac{1}{x}+\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}+...\infty  \right)\] done clear
  D)
   \[2\,\left( \frac{1}{x}-\frac{1}{3{{x}^{3}}}+\frac{1}{5{{x}^{5}}}-...\infty  \right)\] done clear
  View Solution play_arrow

• question_answer21)
  \[{{\log }_{e}}\frac{4}{5}+\frac{1}{4}-\frac{1}{2}{{\left( \frac{1}{4} \right)}^{2}}+\frac{1}{3}\,{{\left( \frac{1}{4} \right)}^{3}}+.....\]
  A)
  \[2{{\log }_{e}}\frac{4}{5}\] done clear
  B)
  \[{{\log }_{e}}\frac{5}{4}\] done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer22)
  \[\frac{1}{{{n}^{2}}}+\frac{1}{2{{n}^{4}}}+\frac{1}{3{{n}^{6}}}+......\infty =\]
  A)
  \[{{\log }_{e}}\left( \frac{{{n}^{2}}}{{{n}^{2}}+1} \right)\] done clear
  B)
  \[{{\log }_{e}}\left( \frac{{{n}^{2}}+1}{{{n}^{2}}} \right)\] done clear
  C)
  \[{{\log }_{e}}\left( \frac{{{n}^{2}}}{{{n}^{2}}-1} \right)\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer23)
  \[\frac{m-n}{m+n}+\frac{1}{3}{{\left( \frac{m-n}{m+n} \right)}^{3}}+\frac{1}{5}{{\left( \frac{m-n}{m+n} \right)}^{5}}+......\infty =\]
  A)
  \[{{\log }_{e}}\left( \frac{m}{n} \right)\] done clear
  B)
  \[{{\log }_{e}}\left( \frac{n}{m} \right)\] done clear
  C)
   \[{{\log }_{e}}\left( \frac{m-n}{m+n} \right)\] done clear
  D)
  \[\frac{1}{2}{{\log }_{e}}\left( \frac{m}{n} \right)\] done clear
  View Solution play_arrow

• question_answer24)
  The sum of \[\frac{1}{2}+\frac{1}{3}.\frac{1}{{{2}^{3}}}+\frac{1}{5}.\frac{1}{{{2}^{5}}}+.....\infty \] is [MP PET 1991]
  A)
  \[{{\log }_{e}}\sqrt{\frac{3}{2}}\] done clear
  B)
  \[{{\log }_{e}}\sqrt{3}\] done clear
  C)
  \[{{\log }_{e}}\sqrt{\frac{1}{2}}\] done clear
  D)
  \[{{\log }_{e}}3\] done clear
  View Solution play_arrow

• question_answer25)
  If   \[4\,\left[ {{x}^{2}}+\frac{{{x}^{6}}}{3}+\frac{{{x}^{10}}}{5}+..... \right]={{y}^{2}}+\frac{{{y}^{4}}}{2}+\frac{{{y}^{6}}}{3}+......,\]then
  A)
  \[{{x}^{2}}y=2x-y\] done clear
  B)
  \[{{x}^{2}}y=2x+y\] done clear
  C)
  \[x=2{{y}^{2}}-1\] done clear
  D)
  \[{{x}^{2}}y=2x+{{y}^{2}}\] done clear
  View Solution play_arrow

• question_answer26)
  \[{{\log }_{a}}x\] is defined for \[(a>0)\] [Roorkee 1990]
  A)
  All real x done clear
  B)
  All negative real \[x\ne 1\] done clear
  C)
  All positive real \[x\ne 0\] done clear
  D)
  \[a\ge e\] done clear
  View Solution play_arrow

• question_answer27)
  \[{{\log }_{e}}2+{{\log }_{e}}\left( 1+\frac{1}{2} \right)+{{\log }_{e}}\left( 1+\frac{1}{3} \right)+....+{{\log }_{e}}\left( 1+\frac{1}{n-1} \right)\] is equal to
  A)
  \[{{\log }_{e}}1\] done clear
  B)
  \[{{\log }_{e}}n\] done clear
  C)
  \[{{\log }_{e}}(1+n)\] done clear
  D)
  \[{{\log }_{e}}(1-n)\] done clear
  View Solution play_arrow

• question_answer28)
  The sum to infinity of the given series \[\frac{1}{n}-\frac{1}{2{{n}^{2}}}+\frac{1}{3{{n}^{3}}}-\frac{1}{4{{n}^{4}}}+....\]  is [MP PET 1994]
  A)
  \[{{\log }_{e}}\left( \frac{n+1}{n} \right)\] done clear
  B)
  \[{{\log }_{e}}\left( \frac{n}{n+1} \right)\] done clear
  C)
  \[{{\log }_{e}}\left( \frac{n-1}{n} \right)\] done clear
  D)
  \[{{\log }_{e}}\left( \frac{n}{n-1} \right)\] done clear
  View Solution play_arrow

• question_answer29)
  The sum of the series \[{{\log }_{4}}2-{{\log }_{8}}2+{{\log }_{16}}2....\] is    [MNR 1994; Roorkee 1994; MP PET 2000]
  A)
  \[{{e}^{2}}\] done clear
  B)
  \[{{\log }_{e}}2\] done clear
  C)
  \[{{\log }_{e}}3-2\] done clear
  D)
  \[1-{{\log }_{e}}2\] done clear
  View Solution play_arrow

• question_answer30)
  The value of \[{{\log }_{3}}e-{{\log }_{9}}e+{{\log }_{27}}e....\] is equal to
  A)
  \[{{\log }_{3}}2\] done clear
  B)
  \[{{\log }_{2}}3\] done clear
  C)
  \[2{{\log }_{3}}2\] v done clear
  D)
  None of the se done clear
  View Solution play_arrow

• question_answer31)
  \[(0.5)-\frac{{{(0.5)}^{2}}}{2}+\frac{{{(0.5)}^{3}}}{3}-\frac{{{(0.5)}^{4}}}{4}+....\] [MP PET 1995]
  A)
  \[{{\log }_{e}}\frac{3}{2}\] done clear
  B)
  \[{{\log }_{10}}\frac{1}{2}\] done clear
  C)
  \[{{\log }_{e}}n\,!\] done clear
  D)
  \[{{\log }_{e}}\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer32)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[{{\log }_{a}}(1+x)\]  is
  A)
  \[\frac{{{(-1)}^{n-1}}}{n}\] done clear
  B)
  \[\frac{{{(-1)}^{n-1}}}{n}{{\log }_{a}}e\] done clear
  C)
  \[\frac{{{(-1)}^{n-1}}}{n}{{\log }_{e}}a\] done clear
  D)
  \[\frac{{{(-1)}^{n}}}{n}{{\log }_{a}}e\] done clear
  View Solution play_arrow

• question_answer33)
  The coefficient of \[{{n}^{-r}}\]in the expansion of \[{{\log }_{10}}\left( \frac{n}{n-1} \right)\] is
  A)
  \[\frac{1}{r\,{{\log }_{e}}10}\] done clear
  B)
  \[-\frac{1}{r{{\log }_{e}}10}\] done clear
  C)
  \[-\frac{1}{r!{{\log }_{e}}10}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[{{\log }_{e}}(1+3x+2{{x}^{2}})\] is [UPSEAT 2001]
  A)
  \[{{(-1)}^{n}}\left[ \frac{{{2}^{n}}+1}{n} \right]\] done clear
  B)
  \[\frac{{{(-1)}^{n+1}}}{n}[{{2}^{n}}+1]\] done clear
  C)
  \[\frac{{{2}^{n}}+1}{n}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  In \[n=(1999)\,!\] then \[\sum\limits_{x=1}^{1999}{{{\log }_{n}}x}\] is equal to  [AMU 2002]
  A)
  1 done clear
  B)
  0 done clear
  C)
  \[\sqrt[1999]{1999}\] done clear
  D)
  - 1 done clear
  View Solution play_arrow

• question_answer36)
  \[{{e}^{\left( x\,-\,\frac{1}{2}{{(x\,-\,1)}^{2}}\,+\,\frac{1}{3}{{(x\,-\,1)}^{3}}\,-\,\frac{1}{4}{{(x\,-\,1)}^{4}}+....... \right)}}\]  is equal to [DCE 2001]
  A)
  \[\log x\] done clear
  B)
  \[\log (x-1)\] done clear
  C)
  x done clear
  D)
  None of these done clear
  View Solution play_arrow