"why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums

The objective is to state why the value of converging alternating seies with terms that are non increasing in magnitude lie between any two consecutive terms of partial sums.

Let alternating series

Sn = partial sum of the series up to  n terms

{S2k} = sequence of partial sum of even terms

{S2k+1} = sequence of partial sum of odd terms

As the magnitude of the terms in the alternating series are non-increasing in magnitude, sequence {S2k} is bounded above by S1 and sequence {S2k+1} is bounded by S2. So, l lies between S1 and S2.

In the similar war, if first two terms of the series are deleted, then l lies in between S3 and S4 and so on.

Hence, the value of converging alternating series with terms that are non-increasing in magnitude lies between any two consecutive terms of partial sums. So, the remainder Rn = S – Sn alternating sign

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