Using the Alternating Series Test, we require two things 1.) lim{n-->oo} a_n=0 and 2.) a_n is a decreasing sequence, for the series to converge. Otherwise, it diverges.
It is obvious that lim{n-->oo} (1/sqrt(n))=0, so condition 1.) is satisfied.
For 2.), consider the function f(x)=1/sqrt(x)
f ' (x)= -1/2*x^(-3/2)<0, for all x>0. Therefore, f(x) is a decreasing sequence. Condition 2.) is also satisfied. Therefore, by the Alternating Series Test, the series converges.
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Using the Alternating Series Test, we require two things 1.) lim{n-->oo} a_n=0 and 2.) a_n is a decreasing sequence, for the series to converge. Otherwise, it diverges.
It is obvious that lim{n-->oo} (1/sqrt(n))=0, so condition 1.) is satisfied.
For 2.), consider the function f(x)=1/sqrt(x)
f ' (x)= -1/2*x^(-3/2)<0, for all x>0. Therefore, f(x) is a decreasing sequence. Condition 2.) is also satisfied. Therefore, by the Alternating Series Test, the series converges.