Let , and . (a) Show that S_2(n) [S_1(n)]^ 2 = 2(2n+1) 3n(n+1) . (b) Hence or…

MATHEMATICS · Subjective · ISC · Mathematics · Algebra · Algebra

Problem

Let

S_1(n)=\sum_{k=1}^{n} k

S_2(n)=\sum_{k=1}^{n} k^{2}

and

S_3(n)=\sum_{k=1}^{n} k^{3}

. (a) Show that

\frac{S_2(n)}{[S_1(n)]^{2}}=\frac{2(2n+1)}{3n(n+1)}.

(b) Hence or otherwise, determine the limiting value of

\dfrac{S_2(n)}{[S_1(n)]^{2}}

n

becomes very large. (c) Using the result in part (b), briefly explain which of the sums

S_1(n)

and

S_2(n)

grows faster for large

n

and why.