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51.

The value of `log 2+2 (1/5+1/3.(1)/(5^(3))+1/5.(1_)/(5^(5))+..+infty)` isA. log2+log3B. log 2+2C. `1/2 log 2`D. log 3

Answer» Correct Answer - d
Discussion
52.

If `S_(n)` denotes the sum of the products of the products of the first n natureal number taken two at a time then `Sigma_(n=0)^(oo) (S_(n))/(n+1!)` equalsA. `(11e)/(24)`B. `(11e)/(12)`C. `(13e)/(24)`D. none of these

Answer» Correct Answer - b
Discussion
53.

`1/(1.2)+(1.3)/(1.2.3.4)+(1.3.5)/(1.2.3.4.5.6)+.....oo`A. `e-1`B. `e^(1//2)-1`C. `e^(1//2)+e`D. none of these

Answer» Correct Answer - d
Discussion
54.

If `a=Sigma_(n=0)^(oo) (x^(3x))/(3n)!,b=Sigma_(n=1)^(oo)(x^(3n-2))/(3n-2!) and C= Sigma_(n=1)^(oo)(x^(3n-1))/(3n-1!)` then the value of `a^(3)+b^(3)+C^(3)-3abc` isA. 1B. 0C. -1D. -2

Answer» Correct Answer - a
Discussion
55.

The sum of the series `(12)/(2!)+(28)/(3!)+(50)/(4!)+(78)/(5!)+`…isA. eB. 3eC. 4eD. 5e

Answer» Correct Answer - a
Discussion
56.

If `S=Sigma_(n=2)^(oo) ""^(n)C_(2) (3^(n-2))/(n!)` then S equalsA. `e^(3//2)`B. `1/2e^(3)`C. `e^(-3//2)`D. `e^(-3)`

Answer» Correct Answer - a
Discussion
57.

If `S=Sigma_(n=0)^(oo) (logx)^(2n)/(2n!)` , then S equalsA. `x+x^(-1)`B. `x-x^(-1)`C. `1/2(x+x^(1))`D. none of these

Answer» Correct Answer - c
Discussion
58.

The sum of the series `S=Sigma_(n=1)^(infty)(1)/(n-1)!` isA. `1//e`B. `e^(2)`C. `-e^(2)`D. e

Answer» Answer:
We have
`S=underset(n=1)overset(infty)Sigma (1)/(n-1)!=1+(1)/(1!)+(1)/(2!)+..infty=e`
Discussion
59.

The sum of the series `5/(1. 2.3)+7/(3. 4.5)+9/(5. 6.7)+` ....is equal toA. `log(8//e)`B. `log(e//8)`C. log 8eD. log 8

Answer» Answer:
We have
`(5)/(1.2.3)+(7)/(3.4.5)+(9)/(5.6.7)`+…
`=underset(n=1)overset(infty)Sigma((2n+3)/(2n-1)(2n)(2n+1))`
Let `(2n+3)/((2n-1)(2n)(2n+1)=(A)/(2n-1)+(B)/(2n)+(C )/(2n+1)`.Then
`A=(1+3)/(1xx(1+1)=2`
`B=(0+3)/((0-1)(0+1))=-3`
`C=(-1+3)/((-1)-1)(-1)=1`
`therefore (2n+3)/((2n-1)(2n)(2n+1))=2((1)/(2n-1)-(1)/(2n))-((1)/(2n)-(1)/(2n+1))`
`rarr underset(n=1)overset(infty)Sigma(2n+3)/((2n-1)(2n)(2n+1))`
`=2log 2 + log2-1=3 log2-loge=log(8//e)`
Discussion
60.

The sum of the series `(1)/(1.23)+(1)/(3.45)+(1)/(5.67)+…infty` isA. `log_(e )2-1/2`B. `log_(e)2`C. `log_(e )2+1/2`D. `log_(e )2+1`

Answer» Answer:
We have
`(1)/(1.2.3)+(1)/(3.4.5)+(1)/(5.6.7)+…infty`
`=underset(n=1)overset(infty)Sigma(1)/((2n-1)2n(2n+1))`
`underset(n=1)overset(infty)Sigma{(1)/(2(2n-1))-(1)/(2n) (1)/(2(2n+1))}`
`=1/2underset(n=1)overset(infty)Sigma{(1)/(2n-1)-(2)/(2n)+(1)/(2n+1)}`
`=1/2 underset(n=1)overset(infty)Sigma{((1)/(2n-1)-(1)/(2n))-((1)/(2n)-(1)/(2n+1))}`
`=1/2log_(e)2+1/2log_(e)2-1/2=log_(e)2-1/2`
Discussion
61.

The sum of the series `1+(1^2+2^2)/(2!)+(1^(2)+2^(2)+3^(2))/(3!)+(1^(2)+2^(2)+3^(2)+4^2)/(4!)`+.. IsA. 3eB. `(17)/(6)e`C. `(13)/(6)e`D. `(13)/(6)e`

Answer» Correct Answer - c
Discussion
62.

The sum of the series `1+(1+2)/(2!)+(1+2+2^(2))/(3!)+(1+2+2^(2)+2^(3))`+…isA. `e^(2)`B. `e^(2)+e`C. `e^(2)-e`D. `e^(2)-e-1`

Answer» Correct Answer - b
Discussion
63.

If `alpha,beta` are the roots of the equation `ax^(2)+bx+c=0` then `log(a-bx+cx^(2))` is equal toA. `log a+(alpha+beta)x+(alpha^(2)+beta^(1))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)`+...B. `log a +(alpha+beta) x+(alpha^(2)+beta^(2))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)`+…C. `log a-(alpha+beta)x-(alpha^(2)+beta^(1))/(2)x^(2)-(alpha^(3)+beta^(3))/(3)x^(3)`+...D. none of these

Answer» Correct Answer - c
Discussion