ISI2015-MMA-24

Question

The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to

• A. $-1$
• B. $1$
• C. $0$
• D. does not converge

Answer 1

Answer:(B)

$\displaystyle\sum _{k=2}^{\infty}\frac{1}{k(k-1)}$

$\displaystyle=\sum _{k=2}^{\infty}\frac{k-(k-1)}{k(k-1)}$

$\displaystyle=\sum _{k=2}^{\infty}\left(\frac{1}{(k-1)}-\frac{1}{k}\right)$

$=\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\ldots$

$=1-\frac{1}{\infty}$

$=1$

Answer 2

1/(k*(k-1))=1/k -  1/(k-1))
Answer is (B) 1