The alternative series 1 - 1/2 + 1/3 - 1/4 + ....... is absolutely : (i) Divergent   (ii) Convergent   (iii) Oscillatory   (iv) None of these 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.

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