1−21+31−41+...+n(−1)n−1+...(∗)an=n(−1)n−1=n21(−1)n−1.
Consider a series of modules
1+21+31+41+...+n1+...(∗∗)
Series
n=1∑∞nα1
convergent if α>1 , divergent if α≤1 .
In our case α=21<1 then series (**) is divergent.
We examine the series (*) on the basis of Leibniz
1)∣a1∣>∣a2∣>...>∣an∣>...
in our case
1>21>31>...>n1>...2)n↦∞lim∣an∣=n↦∞limn1=0.
According to Leibniz, the series (*) is convergent.
Since the series (**) is divergent and the series (*) is convergent, the series (*) is conventionally convergent.
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