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Q. Let $a _{ n }=\int\limits_{-1}^{ n }\left(1+\frac{ x }{2}+\frac{ x ^2}{2}+\frac{ x ^3}{3}+\ldots \ldots . .+\frac{ x ^{ n -1}}{ n }\right) dx$ for $n \in N$. Then the sum of all the elements of the set $\left\{ n \in N : a _{ n } \in(2,30)\right\}$ is ______

JEE MainJEE Main 2022Integrals

Solution:

$ \int\limits_{-1}^{ n }\left(1+\frac{ x }{2}+\frac{ x ^2}{3}+\ldots+\frac{ x ^{ n -1}}{ n }\right) dx $
${\left[ x +\frac{ x ^2}{2}+\frac{ x ^3}{3^2}+\ldots+\frac{ x ^{ n }}{ n ^2}\right]_{-1}^{ n }} $
$\left( n +\frac{ n ^2}{2^2}+\frac{ n ^3}{3^2}+\ldots+-\frac{ n ^{ n }}{ n ^2}\right) $
$ -\left(-1+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^2}+\ldots+\frac{(-1)^{ n }}{ n ^2}\right) $
$ a _{ n }=( n +1)+\frac{1}{2^2}\left( n ^2-1\right)+\frac{1}{3^2}\left( n ^3+1\right) $
$+\ldots+\frac{1}{ n ^2}\left( n ^{ n }-(-1)^{ n }\right)$
if $ n =1 \Rightarrow a _{ n }=2 \notin(2,30)$
if $n =2 $
$ \Rightarrow a _{ n }=(2+1)+\frac{1}{2^2}\left(2^2-1\right)=3+\frac{3}{4}< 30 $
if $ n =3 $
$ \Rightarrow a _{ n }=(3+1)+\frac{1}{4}(8)+\frac{1}{9}(28)=11+\frac{28}{9}< 30$
If $ n =4 $
$ \Rightarrow a _{ n }=(4+1)+\frac{1}{4}(16-1)+\frac{1}{9}(64+1)+\frac{1}{16} $
$ =5+\frac{15}{4}+\frac{65}{9}+\frac{255}{16}>30$
Test $\{2,3\}$ sum of elements 5