-
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
View Solution play_arrow
-
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
View Solution play_arrow
-
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
View Solution play_arrow
-
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
View Solution play_arrow