JEE Main & Advanced Mathematics Sequence & Series Question Bank

done Exponential series Total Questions - 60

• question_answer1)
  If   \[y=1+\frac{x}{1\,!}+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+......\infty \],  then \[x=\]
  A)
  \[{{\log }_{e}}y\] done clear
  B)
  \[{{\log }_{e}}\frac{1}{y}\] done clear
  C)
  \[{{e}^{y}}\] done clear
  D)
  \[{{e}^{-y}}\]! done clear
  View Solution play_arrow

• question_answer2)
  \[1+\frac{1+3}{2\,!}+\frac{1+3+5}{3\,!}+\frac{1+3+5+7}{4\,!}+.......\infty =\]
  A)
  \[e/2\] done clear
  B)
  \[e\] done clear
  C)
  \[2\,e\] done clear
  D)
  \[3e\] done clear
  View Solution play_arrow

• question_answer3)
  \[\frac{1\,.\,2}{1\,!}+\frac{2\,.\,3}{2\,!}+\frac{3\,.\,4}{3\,!}+\frac{4\,.\,5}{4\,!}+.....\infty =\]
  A)
  \[2\,e\] done clear
  B)
  \[3\,e\] done clear
  C)
  \[3\,e-1\] done clear
  D)
  \[e\] done clear
  View Solution play_arrow

• question_answer4)
  The coefficient of  \[{{x}^{r}}\] in the expansion of \[1+\frac{a+bx}{1\,!}+\frac{{{(a+bx)}^{2}}}{2\,!}+.....+\frac{{{(a+bx)}^{n}}}{n\,!}+.....\] is [MP PET 1989]
  A)
  \[\frac{{{(a+b)}^{r}}}{r\,!}\] done clear
  B)
  \[\frac{{{b}^{r}}}{r\,!}\] done clear
  C)
  \[\frac{{{e}^{a}}{{b}^{r}}}{r\,!}\] done clear
  D)
  \[{{e}^{a+{{b}^{r}}}}\] done clear
  View Solution play_arrow

• question_answer5)
  If  \[y=x-\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3!}-\frac{{{x}^{4}}}{4\,!}+......,\] then \[x=\]
  A)
  \[{{\log }_{e}}(1-y)\] done clear
  B)
  \[\frac{1}{{{\log }_{e}}(1-y)}\] done clear
  C)
  \[{{\log }_{e}}\frac{1}{1-y}\] done clear
  D)
  \[{{\log }_{e}}(1+y)\] done clear
  View Solution play_arrow

• question_answer6)
  In  the expansion of  \[\frac{{{e}^{5x}}+{{e}^{x}}}{{{e}^{3x}}}\], the coefficient of  \[{{x}^{4}}\]is
  A)
  - \[6/5\] done clear
  B)
  4/3 done clear
  C)
  - 4/3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  In  the expansion of \[(1+x+{{x}^{2}}){{e}^{-x}}\],  the coefficient of  \[{{x}^{2}}\] is
  A)
  1 done clear
  B)
  \[-1\] done clear
  C)
  1/2 done clear
  D)
  -1/2 done clear
  View Solution play_arrow

• question_answer8)
  In  the expansion of  \[\frac{{{e}^{7x}}+{{e}^{3x}}}{{{e}^{5x}}}\] , the constant term is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  \[\frac{2}{1\,!}+\frac{2+4}{2\,!}+\frac{2+4+6}{3\,!}+....\infty =\]  [MNR 1985]
  A)
  \[e\] done clear
  B)
  \[2\,e\] done clear
  C)
  \[3\,e\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  \[{{\left[ 1+\frac{1}{2\,!}+\frac{1}{4\,!}+....\infty  \right]}^{2}}-{{\left[ 1+\frac{1}{3\,!}+\frac{1}{5\,!}+.....\infty  \right]}^{2}}=\]
  A)
   0 done clear
  B)
  1 done clear
  C)
  \[-1\] done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer11)
  \[1+\frac{1}{3\,!}+\frac{1}{5\,!}+\frac{1}{7\,!}+.....\infty =\] [MP PET 1991]
  A)
  \[{{e}^{-1}}\] done clear
  B)
  \[e\] done clear
  C)
  \[\frac{e+{{e}^{-1}}}{2}\] done clear
  D)
  \[\frac{e-{{e}^{-1}}}{2}\] done clear
  View Solution play_arrow

• question_answer12)
  In  the expansion of \[{{({{e}^{x}}-1)}^{2}}\], the coefficient of \[{{x}^{4}}\] will be
  A)
  1/12 done clear
  B)
  7/12 done clear
  C)
  5/12 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  \[1+\frac{{{2}^{3}}}{2\,!}+\frac{{{3}^{3}}}{3\,!}+\frac{{{4}^{3}}}{4\,!}+....\infty \] =  [MNR 1976; MP PET 1997]
  A)
  \[2\,e\] done clear
  B)
  \[3\,e\] done clear
  C)
  \[4\,e\] done clear
  D)
  \[5\,e\] done clear
  View Solution play_arrow

• question_answer14)
  \[\frac{2}{3\,!}+\frac{4}{5\,!}+\frac{6}{7\,!}+......\infty =\]   [MNR 1979; MP PET 1995, 2002; Pb. CET 2002]
  A)
  \[e\] done clear
  B)
  \[2\,e\] done clear
  C)
  \[{{e}^{2}}\] done clear
  D)
  \[1/e\] done clear
  View Solution play_arrow

• question_answer15)
  \[\frac{{{x}^{2}}-{{y}^{2}}}{1\,!}+\frac{{{x}^{4}}-{{y}^{4}}}{2\,!}+\frac{{{x}^{6}}-{{y}^{6}}}{3\,!}+......\infty =\]
  A)
  \[{{e}^{x}}-{{e}^{y}}\] done clear
  B)
  \[{{e}^{{{x}^{2}}}}-{{e}^{{{y}^{2}}}}\] done clear
  C)
  \[2+{{e}^{{{x}^{2}}}}-{{e}^{{{y}^{2}}}}\] done clear
  D)
  \[\frac{{{e}^{x}}-{{e}^{y}}}{2}\] done clear
  View Solution play_arrow

• question_answer16)
  \[1+\frac{a-b}{a}+\frac{1}{2\,!}{{\left( \frac{a-b}{a} \right)}^{2}}+\frac{1}{3\,!}{{\left( \frac{a-b}{a} \right)}^{3}}+......\infty =\]
  A)
  \[x=\] done clear
  B)
  \[{{e}^{a}}\] done clear
  C)
  \[\frac{e}{{{e}^{b/a}}}\] done clear
  D)
  \[\frac{e}{{{e}^{a/b}}}\] done clear
  View Solution play_arrow

• question_answer17)
  \[3+\frac{5}{1\,!}+\frac{7}{2\,!}+\frac{9}{3\,!}+.....\infty =\]
  A)
  \[3\,e\] done clear
  B)
  \[5\,e\] done clear
  C)
  \[5\,e-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  In  the expansion of  \[\frac{{{e}^{4x}}-1}{{{e}^{2x}}}\], the coefficient of \[{{x}^{2}}\] is
  A)
  \[\frac{1}{2}\] done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  \[1+\frac{a-bx}{1\,!}+\frac{{{(a-bx)}^{2}}}{2\,!}+\frac{{{(a-bx)}^{3}}}{3\,!}+....\infty =\]
  A)
  \[{{e}^{a-bx}}\] done clear
  B)
  \[{{e}^{a-bx}}-1\] done clear
  C)
  \[1+a{{\log }_{e}}(a-bx)\] done clear
  D)
  \[{{e}^{-bx}}\] done clear
  View Solution play_arrow

• question_answer20)
  If  \[y=-\left( {{x}^{3}}+\frac{{{x}^{6}}}{2}+\frac{{{x}^{9}}}{3}+..... \right)\], then \[x=\] [MNR 1975]
  A)
  \[\frac{1+{{e}^{y}}}{3}\] done clear
  B)
  \[\frac{1-{{e}^{y}}}{3}\] done clear
  C)
  \[{{(1-{{e}^{y}})}^{1/3}}\] done clear
  D)
  \[{{(1-{{e}^{y}})}^{3}}\] done clear
  View Solution play_arrow

• question_answer21)
  \[\frac{{{e}^{2}}+1}{2\,e}=\]
  A)
  \[1+\frac{2}{2\,!}+\frac{{{2}^{2}}}{3\,!}+\frac{{{2}^{3}}}{4\,!}+.....\infty \] done clear
  B)
   \[1+\frac{1}{2\,!}+\frac{1}{4\,!}+\frac{1}{6\,!}+.....\infty \] done clear
  C)
  \[\frac{1}{2}\left( 1+\frac{1}{2\,!}+\frac{1}{4\,!}+....\infty  \right)\] done clear
  D)
  \[\frac{1}{2}\left( 1+\frac{1}{1\,!}+\frac{1}{2\,!}+\frac{1}{3\,!}+....\infty  \right)\] done clear
  View Solution play_arrow

• question_answer22)
  \[\left( 1+\frac{1}{2\,!}+\frac{1}{4\,!}+.... \right)\,\,\left( 1+\frac{1}{3\,!}+\frac{1}{5\,!}+.... \right)\,=\]
  A)
  \[{{e}^{4}}\] done clear
  B)
  \[\frac{{{e}^{2}}-1}{{{e}^{2}}}\] done clear
  C)
  \[\frac{{{e}^{4}}-1}{4\,{{e}^{2}}}\] done clear
  D)
  \[\frac{{{e}^{4}}+1}{4\,{{e}^{2}}}\] done clear
  View Solution play_arrow

• question_answer23)
  \[\frac{{{1}^{2}}.2}{1\,!}+\frac{{{2}^{2}}.3}{2\,!}+\frac{{{3}^{2}}.4}{3\,!}+.....\infty =\] [UPSEAT 1999]
  A)
  \[6\,e\] done clear
  B)
  \[7\,e\] done clear
  C)
  \[8\,e\] done clear
  D)
  \[9\,e\] done clear
  View Solution play_arrow

• question_answer24)
  \[\frac{1}{2}+\frac{1}{4}+\frac{1}{8(2)\,!}+\frac{1}{16\,(3)\,!}+\frac{1}{32(4)\,!}+......\infty =\]
  A)
  \[e\] done clear
  B)
  \[\sqrt{e}\] done clear
  C)
  \[\frac{\sqrt{e}}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  The value of \[\frac{2\frac{1}{2}}{1\,!}+\frac{3\frac{1}{2}}{2\,!}+\frac{4\frac{1}{2}}{3\,!}+\frac{5\frac{1}{2}}{4\,!}+......\infty \] is
  A)
  \[1+e\] done clear
  B)
  \[\frac{1+e}{e}\] done clear
  C)
  \[\frac{e-1}{e}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  The coefficient of \[{{x}^{r}}\] in the expansion of \[{{e}^{{{e}^{x}}}}\] is
  A)
  \[\frac{{{1}^{r}}}{1\,!}+\frac{{{2}^{r}}}{2\,!}+\frac{{{3}^{r}}}{3\,!}+.....\] done clear
  B)
  \[1+\frac{1}{1\,!}+\frac{1}{2\,!}+....+\frac{1}{r\,!}\] done clear
  C)
  \[\frac{1}{r\,!}\left[ \frac{{{1}^{r}}}{1\,!}+\frac{{{2}^{r}}}{2\,!}+\frac{{{3}^{r}}}{3\,!}+.... \right]\] done clear
  D)
  \[\frac{{{e}^{r}}}{r!}\] done clear
  View Solution play_arrow

• question_answer27)
  If   \[{{T}_{n}}=\frac{{{3}^{n}}}{2\,(n\,!)}-\frac{1}{2\,(n\,!)},\] then \[{{S}_{\infty }}=\]
  A)
  \[\frac{{{e}^{3}}-1}{2}\] done clear
  B)
  \[\frac{{{e}^{3}}-e}{2}\] done clear
  C)
  \[\frac{e-3}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer28)
  Sum to infinity of the series is  \[1+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{4}}}{4\,!}+......\] is [MP PET 1994]
  A)
  \[\frac{{{e}^{x}}-{{e}^{-x}}}{2}\] done clear
  B)
  \[\frac{{{e}^{x}}+{{e}^{-x}}}{2}\] done clear
  C)
  \[\frac{{{e}^{-x}}-{{e}^{x}}}{2}\] done clear
  D)
    \[\frac{-({{e}^{x}}+{{e}^{-x}})}{2}\] done clear
  View Solution play_arrow

• question_answer29)
  Which of the following is not true   [Kurukshetra CEE 1996]
  A)
  \[\log (1+x)<x\] for \[x>0\] done clear
  B)
  \[\frac{x}{1+x}<\log (1+x)\]for \[x>0\] done clear
  C)
  \[{{e}^{x}}>1+x\] for \[x>0\] done clear
  D)
  \[{{e}^{-x}}<1-x\] for \[x>0\] done clear
  View Solution play_arrow

• question_answer30)
  \[1+\frac{{{({{\log }_{e}}n)}^{2}}}{2\,!}+\frac{{{({{\log }_{e}}n)}^{4}}}{4\,!}+....=\] [MP PET 1996]
  A)
  \[n\] done clear
  B)
  \[1/n\] done clear
  C)
  \[\frac{1}{2}(n+{{n}^{-1}})\] done clear
  D)
    \[\frac{1}{2}({{e}^{n}}+{{e}^{-n}})\] done clear
  View Solution play_arrow

• question_answer31)
  \[1+\frac{1+2}{2\,!}+\frac{1+2+3}{3\,!}+\frac{1+2+3+4}{4\,!}+....\infty =\] [Roorkee 1999; MP PET 2003]
  A)
  \[e\] done clear
  B)
  \[3\,e\] done clear
  C)
   \[e/2\] done clear
  D)
    \[3e/2\] done clear
  View Solution play_arrow

• question_answer32)
  The value of \[1-\log 2+\frac{{{(\log 2)}^{2}}}{2\,!}-\frac{{{(\log 2)}^{3}}}{3\,!}+....\]  is   [MP PET 1998; Pb. CET 2000]
  A)
  2 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\log 3\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[\frac{{{e}^{7x}}+{{e}^{x}}}{{{e}^{3x}}}\] is    [MP PET 1999]
  A)
  \[\frac{{{4}^{n-1}}+{{(-2)}^{n}}}{n\,!}\] done clear
  B)
  \[\frac{{{4}^{n-1}}+{{2}^{n}}}{n\,!}\] done clear
  C)
  \[\frac{{{4}^{n-1}}+{{(-2)}^{n-1}}}{n\,!}\] done clear
  D)
  \[\frac{{{4}^{n}}+{{(-2)}^{n}}}{n\,!}\] done clear
  View Solution play_arrow

• question_answer34)
  If \[S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\frac{1}{4.5}+....+\infty ,\] then \[{{e}^{S}}=\]  [MP PET 1999]
  A)
  \[{{\log }_{e}}\left( \frac{4}{e} \right)\] done clear
  B)
  \[\frac{4}{e}\] done clear
  C)
  \[{{\log }_{e}}\left( \frac{e}{4} \right)\] done clear
  D)
  \[\frac{e}{4}\] done clear
  View Solution play_arrow

• question_answer35)
  The value of \[\sqrt{e}\] will be [UPSEAT 1999]
  A)
  1.648 done clear
  B)
  1.547 done clear
  C)
  1.447 done clear
  D)
  1.348 done clear
  View Solution play_arrow

• question_answer36)
  The sum of the infinite series  \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+......\]is  [AMU 1999]
  A)
  \[e-2\] done clear
  B)
  \[\frac{2}{3}e-1\] done clear
  C)
  1 done clear
  D)
  3/2 done clear
  View Solution play_arrow

• question_answer37)
  Sum of the infinite series \[1+2+\frac{1}{2!}+\frac{2}{3!}+\frac{1}{4!}+\frac{2}{5!}+.....\] is [Roorkee 2000]
  A)
  \[{{e}^{2}}\] done clear
  B)
  \[e+{{e}^{-1}}\] done clear
  C)
  \[\frac{e-{{e}^{-1}}}{2}\] done clear
  D)
  \[\frac{3e-{{e}^{-1}}}{2}\] done clear
  View Solution play_arrow

• question_answer38)
  The sum of \[\frac{2}{1\,!}+\frac{6}{2\,!}+\frac{12}{3\,!}+\frac{20}{4\,!}+\].......is [UPSEAT 2000]
  A)
  \[\frac{3e}{2}\] done clear
  B)
  \[e\] done clear
  C)
  \[2e\] done clear
  D)
  \[3e\] done clear
  View Solution play_arrow

• question_answer39)
  The sum of the series \[\frac{1}{2\,!}-\frac{1}{3\,!}+\frac{1}{4\,!}-.....\] is [DCE 2002]
  A)
  e done clear
  B)
  \[{{e}^{-\,\frac{1}{2}}}\] done clear
  C)
  \[{{e}^{-\,2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  The sum of the series\[\frac{1}{1.2}+\frac{1.3}{1.2.3.4}+\frac{1.3.5}{1.2.3.4.5.6}+.....\infty \] is [Kurukshetra CEE 2002]
  A)
  \[15e\] done clear
  B)
  \[{{e}^{1/2}}+e\] done clear
  C)
  \[{{e}^{1/2}}-1\] done clear
  D)
  \[{{e}^{1/2}}-e\] done clear
  View Solution play_arrow

• question_answer41)
  \[1+x{{\log }_{e}}a+\frac{{{x}^{2}}}{2\,!}{{({{\log }_{e}}a)}^{2}}+\frac{{{x}^{3}}}{3\,!}{{({{\log }_{e}}a)}^{3}}+...=\] [EAMCET 2002]
  A)
  \[{{a}^{x}}\] done clear
  B)
  x done clear
  C)
  \[{{a}^{{{\log }_{a}}x}}\] done clear
  D)
  a done clear
  View Solution play_arrow

• question_answer42)
  \[\frac{1+\frac{{{2}^{2}}}{2\,!}+\frac{{{2}^{4}}}{3\,!}+\frac{{{2}^{6}}}{4\,!}+.....\infty }{1+\frac{1}{2\,!}+\frac{2}{3\,!}+\frac{{{2}^{2}}}{4\,!}+....\infty }=\]
  A)
  \[{{e}^{2}}\] done clear
  B)
  \[{{e}^{2}}-1\] done clear
  C)
  \[{{e}^{3/2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  \[1+\frac{{{2}^{4}}}{2\,!}+\frac{{{3}^{4}}}{3\,!}+\frac{{{4}^{4}}}{4\,!}+.....\infty =\]
  A)
  \[5\,e\] done clear
  B)
  \[e\] done clear
  C)
  \[15\,e\] done clear
  D)
  \[2\,e\] done clear
  View Solution play_arrow

• question_answer44)
  In  the expansion of \[({{e}^{x}}-1)\,({{e}^{-x}}+1)\], the coefficient of \[{{x}^{3}}\] is
  A)
  0 done clear
  B)
  1/3 done clear
  C)
  2/3 done clear
  D)
  1/6 done clear
  View Solution play_arrow

• question_answer45)
  \[\frac{2}{1\,!}+\frac{4}{3\,!}+\frac{6}{5\,!}+\frac{8}{7\,!}+......\infty =\]   [JMI CET 2000]
  A)
  \[1/e\] done clear
  B)
  \[e\] done clear
  C)
  \[2\,e\] done clear
  D)
  \[3e\] done clear
  View Solution play_arrow

• question_answer46)
  \[1+\frac{3}{1\,!}+\frac{5}{2\,!}+\frac{7}{3\,!}+....\infty =\]
  A)
  \[e\] done clear
  B)
  \[2\,e\] done clear
  C)
  \[3\,e\] done clear
  D)
  \[4\,e\] done clear
  View Solution play_arrow

• question_answer47)
  \[1-x+\frac{{{x}^{2}}}{2\,!}-\frac{{{x}^{3}}}{3\,!}+....\infty =\] [MP PET 1986]
  A)
  \[{{e}^{x}}\] done clear
  B)
  \[{{e}^{-x}}\] done clear
  C)
  \[e\] done clear
  D)
  \[{{e}^{{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer48)
  \[1+\frac{1+x}{2\,!}+\frac{1+x+{{x}^{2}}}{3\,!}+\frac{1+x+{{x}^{2}}+{{x}^{3}}}{4\,!}+.....\infty =\]
  A)
  \[\frac{{{e}^{x}}+1}{x+1}\] done clear
  B)
  \[\frac{{{e}^{x}}+1}{x-1}\] done clear
  C)
  \[\frac{{{e}^{x}}-e}{x+1}\] done clear
  D)
  \[\frac{{{e}^{x}}-e}{x-1}\] done clear
  View Solution play_arrow

• question_answer49)
  In  the expansion of  \[\frac{a+bx}{{{e}^{x}}}\],  the coefficient of \[{{x}^{r}}\] is
  A)
  \[\frac{a-b}{r\,!}\] done clear
  B)
    \[\frac{a-br}{r\,!}\] done clear
  C)
  \[{{(-1)}^{r}}\frac{a-br}{r\,!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  \[1+\frac{{{4}^{2}}}{3\,!}+\frac{{{4}^{4}}}{5\,!}+......\infty =\]
  A)
  \[\frac{{{e}^{4}}+{{e}^{-4}}}{4}\] done clear
  B)
  \[\frac{{{e}^{4}}-{{e}^{-4}}}{4}\] done clear
  C)
  \[\frac{{{e}^{4}}+{{e}^{-4}}}{8}\] done clear
  D)
  \[\frac{{{e}^{4}}-{{e}^{-4}}}{8}\] done clear
  View Solution play_arrow

• question_answer51)
  \[1+\frac{{{2}^{2}}}{1\,!}+\frac{{{3}^{2}}}{2\,!}+\frac{{{4}^{2}}}{3\,!}+......\infty =\]
  A)
  \[2\,e\] done clear
  B)
  \[3\,e\] done clear
  C)
  \[(0.5)-\frac{{{(0.5)}^{2}}}{2}+\frac{{{(0.5)}^{3}}}{3}-\frac{{{(0.5)}^{4}}}{4}+....\] done clear
  D)
  \[5\,e\] done clear
  View Solution play_arrow

• question_answer52)
  In  the expansion of  \[\frac{1-2x+3{{x}^{2}}}{{{e}^{x}}}\], the coefficient of \[{{x}^{5}}\] will be
  A)
  \[\frac{71}{120}\] done clear
  B)
  \[-\frac{71}{120}\] done clear
  C)
  \[\frac{31}{40}\] done clear
  D)
  \[-\frac{31}{40}\] done clear
  View Solution play_arrow

• question_answer53)
  \[1+\frac{2}{3\,!}+\frac{3}{5\,!}+\frac{4}{7\,!}+......\infty =\,\]
  A)
  e done clear
  B)
  \[2\,e\] done clear
  C)
  e/2 done clear
  D)
  e/3 done clear
  View Solution play_arrow

• question_answer54)
  \[\frac{1}{2\,!}+\frac{1+2}{3\,!}+\frac{1+2+3}{4\,!}+......\infty =\] [EAMCET 2003]
  A)
  \[e\] done clear
  B)
  \[2\,e\] done clear
  C)
  e/2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  If  \[i=\sqrt{-1}\],  then  \[\frac{{{e}^{xi}}+{{e}^{-xi}}}{2}=\]
  A)
  \[1+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{4}}}{4\,!}+.....\infty \] done clear
  B)
  \[1-\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{4}}}{4\,!}-.....\infty \] done clear
  C)
  \[x+\frac{{{x}^{3}}}{3\,!}+\frac{{{x}^{5}}}{5\,!}+....\infty \] done clear
  D)
  \[i\,\left[ x-\frac{{{x}^{3}}}{3\,!}+\frac{{{x}^{5}}}{5\,!}-.....\infty  \right]\] done clear
  View Solution play_arrow

• question_answer56)
  The sum of the series \[\frac{{{1}^{2}}}{1\cdot 2\,!}+\frac{{{1}^{2}}+{{2}^{2}}}{2\cdot 3\,!}+\frac{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}{3\cdot 4\,!}+..+\frac{{{1}^{2}}+{{2}^{2}}+...+{{n}^{2}}}{n\cdot (n+1)\,!}+...\infty \]equals [AMU 2002]
  A)
  \[{{e}^{2}}\] done clear
  B)
  \[\frac{1}{2}{{(e+{{e}^{-1}})}^{2}}\] done clear
  C)
  \[\frac{3e-1}{6}\] done clear
  D)
  \[\frac{4e+1}{6}\] done clear
  View Solution play_arrow

• question_answer57)
  \[\frac{2}{1\,!}{{\log }_{e}}2+\frac{{{2}^{2}}}{2\,!}{{({{\log }_{e}}2)}^{2}}+\frac{{{2}^{3}}}{3\,!}{{({{\log }_{e}}2)}^{3}}+.....\infty =\]
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  None of these done clear
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• question_answer58)
  \[1+\frac{{{\log }_{e}}x}{1\,!}+\frac{{{({{\log }_{e}}x)}^{2}}}{2\,!}+\frac{{{({{\log }_{e}}x)}^{3}}}{3\,!}+.....\infty =\] [Kurukshetra CEE 1998; JMI CET 2000]
  A)
  \[{{\log }_{e}}x\] done clear
  B)
  \[x\] done clear
  C)
  \[{{x}^{-1}}\] done clear
  D)
  \[-{{\log }_{e}}(1+x)\] done clear
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• question_answer59)
  \[(1+3){{\log }_{e}}3+\frac{1+{{3}^{2}}}{2\,!}{{({{\log }_{e}}3)}^{2}}+\frac{1+{{3}^{3}}}{3\,!}{{({{\log }_{e}}3)}^{3}}+.....\infty =\] [Roorkee 1989]
  A)
  28 done clear
  B)
  30 done clear
  C)
  25 done clear
  D)
  0 done clear
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• question_answer60)
  The coefficients of \[{{x}^{3}}\] in the expansion of \[{{3}^{x}}\] is [Kerala (Engg.) 2005]
  A)
  \[\frac{{{3}^{3}}}{6}\] done clear
  B)
  \[\frac{{{(\log 3)}^{3}}}{3}\] done clear
  C)
  \[\frac{\log ({{3}^{3}})}{6}\] done clear
  D)
  \[\frac{{{(\log 3)}^{3}}}{6}\] done clear
  E)
  \[\frac{3}{3\,!}\] done clear
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