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Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ? a2 ? a3 ? ·...

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Advance Math

Alternating Series Test. Let (an) be a sequencesatisfying
(i) a1 ? a2 ? a3 ? · · · ? an ? an+1 ? · · · and
(ii) (an) ? 0.
Show that then the alternating series X?
n=1
(?1)n+1an converges using the following two differentapproaches.
(a) Show that the sequence (sn) of partial sums,
sn = a1 ? a2 + a3 ? · · · ± an
is a Cauchy sequence
Alternating Series Test. Let (an) be a sequencesatisfying
(i) a1 ? a2 ? a3 ? · · · ? an ? an+1 ? · · · and
(ii) (an) ? 0.
Show that then the alternating series X?
n=1
(?1)n+1an converges using the following two differentapproaches.
(a) Show that the sequence (sn) of partial sums,
sn = a1 ? a2 + a3 ? · · · ± an
is a Cauchy sequence.
(b) Consider the subsequences (s2n) and (s2n+1) and use theMonotone Convergence Theorem to show convergence of the originalseries.

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