Question

Consider the telescoping series

$n$

$Sn = Σ$ $[ ( 1 / (k+1) ) - ( 1 / k )$ ]

$k=2$

Which of the options below are incorrect?

1) $Sn = [ (1-n) / ( 2n + 2) )$ ]

2)

$n + 1$

$Sn$ = - $Σ [ 1 / k ( k - 1) ]$

$k = 3$

3)

$n$

$Sn = Σ [ 1 / k ( k + 1) ]$

$k = 2$

4)

$∞$

$Sn = Σ [ 1 / k ( k + 1) ]$ = $1/2$

$k = 2$

5)

$∞$

Sn = $Σ [ 1 / k ( k - 1) ]$ $= - 1/2$

$k = 3$

Answer 1

Given, the telescoping series

$S_n=Σ_{k=2}^{n}(\frac{1}{k+1}-\frac{1}{k})\\ =(\frac{1}{3}-\frac{1}{2})+(\frac{1}{4}-\frac{1}{3})+(\frac{1}{5}-\frac{1}{4}) +•••+(\frac{1}{n+1}-\frac{1}{n})\\ =(\frac{1}{n+1}-\frac{1}{2})\\ =\frac{1-n}{2n+2}\\ \text{Thus, option (1) is correct.}\\ S_n=Σ_{k=2}^{n}(\frac{1}{k+1}-\frac{1}{k})\\ =Σ_{k=2}^{n}\frac{-1}{k(k+1)}\\ =-(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{6.7}+\frac{1}{7.8} +•••+\frac{1}{n(n+1)})\\ =-(\frac{1}{3.2}+\frac{1}{4.3}+\frac{1}{5.4}+\frac{1}{7.6}+\frac{1}{8.7} +•••+\frac{1}{(n+1).n})\\ =Σ_{k=3}^{n+1}\frac{-1}{k(k-1)}\\ \text{Thus, option (2) is correct.}$