Let `s_k, k=1,2,3,…,100` denote the sum of the infinite geometric series whose first term is `(k-1)/|___k` and the common ratio is `1/k`. Then, the value of
is

Let S_(k) , where k = 1,2 ,....,100, denotes the sum of the infinite geometric series whose first term is (k -1)/(k!) and the common ratio is (1)/(k) . Then, the value of (100^(2))/(100!) +sum_(k=2)^(100) | (k^(2) - 3k +1) S_(k)| is....

Let S_(k) , be the sum of an infinite geometric series whose first term is kand common ratio is (k)/(k+1)(kgt0) . Then the value of sum_(k=1)^(oo)(-1)^(k)/(S_(k)) is equal to

Let S_(k) denote sum of infinite geometric series (K=1,2,3,.......), where first term is (k^(2)-1) and common ratio is (1)/(K), then the unit digit of the sum (sum_(k=1)^(oo)(S_(k))/(2^(k-1))), is

Find the sum of an infinite geometric series whose first term is lim_(x rarr0)sum_(k=1)^(2011)(((x)/(tan x)+2k))/(2011) and whose common ratio is the value of lim_(x rarr0)(e^(tan^(3)x)-e^(x^(3)))/(2ln(1+x^(3)sin^(2)x))[ Note: (y) denotes fractional part of yl.

Let S_(K) be the sum of an infinte G.P. series whose first term is K and common ratio is (K)/(K+1)(Kgt0) . Then the value of sum_(K=1)^(oo)((-1)^(K))/(S_(K)) is equal to

If S_1, S_2,S_3,S_4,.....,S_p denotes the sums of infinite geometric series whose first terms are 1, 2, 3,...p respectively and whose common ratios are 1/2,1/3,1/4,.....1/(p+1) respectively .then S_1+S_2+S_3+....+S_p = kp(p+3), where k =?

Let S_k be sum of an indinite G.P whose first term is 'K' and commmon ratio is (1)/(k+1) . Then Sigma_(k=1)^(10) S_k is equal to _________.