determine whether the series converges if so find its sum th

Solution

this series can be written as 2 ( 1/k - 1/(k+2)) T1 = 2 (1/1 - 1/3 ) T2 = 2 ( 1/2 - 1/4) T3 = 2 (1/3 - 1/5) ....... ....... Tn = 2 (1/n - 1/n+2) we can see that term are getting cancelled (2nd term of T1 and 1st term of T3) this process will continue and if we all the terms together we will be left with Sn = 2 (1/1 + 1/2 + 1/n+2 + 1/n+1) putting n = infinity we get Sum = 3