Question

If $S_n=\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+....+\frac{1+2+3+.....+n}{1^3+2^3+3^3+....+n^{3,}}n=1,2,3,....$Then $S_n$ is not greater then

• A. $1/2$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

$Sn=\frac{1}{1^3}+\frac{1+2}{1^3+2^3}+.....\frac{1+2+3+....n}{1^3+2^3+....n^2}$ will be greater than $1$ because its first term is $1$

Now Let us assume series $Tn=1+\frac{1}{2}+\frac{1}{4}....\frac{1}{2^n}=2$ (when $h=\infty$)

So $\sin Sn$ & $Tn$ every term in $Sn<Tn(\frac{1}{2}>\frac{3}{8},\frac{1}{4}>\frac{1}{6})$

$Sn$ is not greater than $2$

$C$ is correct