Let us determine whether the following series converge, converge absolutely, converge conditionally, or diverge.
1) k=1∑∞k22k3k
Let us use D'Alembert's Ratio Test.
Since n→∞lim((k+1)22k+13k+1:k22k3k)=n→∞lim2(k+1)23k2=n→∞lim2(k2+2k+1)3k2=23>1,
we conclude that this series diverge.
2) Since
k=1∑∞k+1−k(−1)k+1=k=1∑∞(k+1−k)(k+1−k)(−1)k+1(k+1+k)=k=1∑∞((k+1)−k)(−1)k+1(k+1+k)=k=1∑∞(−1)k+1(k+1+k),
and k→∞lim(k+1+k)=∞,
we conclude that the necessary condition of covergence is not met, and hence this series is divergent.
3) k=1∑∞(−1)k+1(k+1−k)
Consider the series k=1∑∞∣(−1)k+1(k+1−k)∣=k=1∑∞(k+1−k).
Since An=(2−1)+(3−2)+(4−3)+…+(n+1−n)=n+1−1,
and n→∞limAn=n→∞lim(n+1−1)=∞, we conclude that the series k=1∑∞(−1)k+1(k+1−k) diverge absolutely.
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