Question

Let the sum of the series $\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+.....+\frac{1+2+.....+n}{1^3+2^3+.....+n^3}$ upto $n$ terms be $S_n,n=1,2,3,.....$. Then $S_n$ cannot be greater than

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (C), (D)

The correct options are
C $2$
D $4$

$T_r=\frac{1+2+3+.....+r}{1^3+2^3+.....+r^3}$
$\frac{2}{r(r+1)}=2(\frac{1}{r}-\frac{1}{(r+1)})$
$\Rightarrow S_n=\sum _{r=1}^n{T}_r=2\sum _{r=1}^n(\frac{1}{r}-\frac{1}{r+1})=2(1-\frac{1}{n+1})$
Which cannot be greater than 2.
Hence, options 'C' and 'D' are correct.