Question #8d376

1 Answer
Feb 6, 2017

Answer:

Refer to the Explanation given below for the Answer : # 2017-1/2017.#

Explanation:

Let us observe that the general #n^(th)" term "T_n# of the given

Series is given by,

#T_n^2=1+1/n^2+1/(n+1)^2#

#={n^2(n+1)^2+(n+1)^2+n^2]/{n^2(n+1)^2}#

#=[{n(n+1)}^2+n^2+2n+1+n^2]/{n(n+1)}^2#

#=[{n(n+1)}^2+2n^2+2n+1]/{n(n+1)}^2#

#=[{n(n+1)}^2+2(1){n(n+1)}+1^2]/[{n(n+1)}^2}#

#={n(n+1)+1}^2/[{n(n+1)}^2}#

#=[{n(n+1)+1}/{n(n+1)}]^2#

#rArr T_n={n(n+1)+1}/{n(n+1)}={n(n+1))/{n(n+1))+1/{n(n+1)}#

#rArr T_n=1+{(n+1)-n}/{n(n+1)}=1+cancel(n+1)/{n(cancel(n+1))}-canceln/{canceln(n+1)}#

#:. T_n=1+1/n-1/(n+1)#

Hence, #S_n=sumT_n=sum[1+1/n-1/(n+1)]#

#=sum1+sum{1/n-1/(n+1)}=n+sum{1/n-1/(n+1)}#

#=n+{(1/1cancel(-1/2))+(cancel(1/2)cancel(-1/3))+(cancel(1/3)cancel(-1/4))+...+(cancel(1/(n-1))cancel(-1/n))+(cancel(1/n)-1/(n+1)}#

#"So, in general, "S_n=n+{1-1/(n+1)}=(n+1)-1/((n+1)).#

In particular, #S_2016=2017-1/2017.#

Enjoy Maths,, and Spread its Joy!