The alternative series 1 - 1/2 + 1/3 - 1/4 + ....... is absolutely : (i) Divergent (ii) Convergent (iii) Oscillatory (iv) None of these
You should perform the alternating series test, hence, if the series `sum a_n, a_n = (-1)^(n+1)*b_n, lim_(n->oo)b_n =0` and the sequence `b_n` decreases, then the series sum `a_n` converges.
You should write the given alternating series such that:
`1 - 1/2 + 1/3 - 1/4 + .... = sum_(k=1)^n ((-1)^(n+1))/n`
`sum_(k=1)^n ((-1)^(n+1))/n = sum_(k=1)^n ((-1)^(n+1))*(1/n)`
Notice that you should note `b_n = 1/n` and you need to test if `lim_(n->oo)b_n = 0` such that:
`lim_(n->oo)(1/n) = 1/oo = 0`
Notice that `b_n = 1/n` and `b_(n+1) = 1/(n+1)` , hence 1/n > 1/(n+1) , thus, the sequence is decreasing.
Since the conditions of convergence are satisfied, the given alternating series is convergent.