Question

Give an example of a conditionally convergent series with sum zero. Prove
that your series is convergent.

Answer 1

An example of a conditionally convergent series with sum zero is 1−3/4+1−1+1/4−3/8+1/2−1/2+1/8−3/16+1/3−1/3+1/16 .....

. A series is convergent if the sequence of its sums tends to a limit, That means the sum becomes closer and closer to a specific number or converges or shrinks to a number.

. Given a series Σ$a_{n}$  whose limits are from n = 1 to n = ∞. If Σ$a_{n}$ n = 1 to n = ∞

converges, but the corresponding series ΣI$a_{n}$I do not converge then

 Σ$a_{n}$ n = 1 to n = ∞ converges conditionally.

. The given series i.e. 1−3/4+1−1+1/4−3/8+1/2−1/2+1/8−3/16+1/3−1/3+1/16 is convergent as it satisfies the alternating series test.

. Above is an example of the Alternating Harmonic Series. The underlying sequence is  $a_{n}$ , which is positive, decreasing, and approaches 0 as  n→ ∞ . Therefore we can apply the Alternating Series Test and come to the conclusion that the given example of series is convergent.

Hence proved.

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