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

Alternating Series Test. Let (an) be a sequence satisfying

(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 different approaches.

(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 sequence satisfying

(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 different approaches.

(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 the Monotone Convergence Theorem to show convergence of the original series.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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