The alternating harmonic series S= _ n=1 ^ (-1)^ n+1 n =1- 12+ 13- 14+ 15- can be…
MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
The alternating harmonic series
S=n=1∑∞n(−1)n+1=1−21+31−41+51−⋯
can be rearranged in many different ways. Two specific rearrangements are described below.
Rearrangement I: Pair consecutive terms and write each pair as a single quantity, yielding
S1=(1−21)+(31−41)+(51−61)+⋯.
Rearrangement II: Group two positive terms followed by one negative term:
S2=(1+31)−21+(51+71)−41+(91+111)−61+⋯.
(a) Verify that the original series S is convergent but not absolutely convergent.
(b) Determine whether the series S1 converges, and if it does, decide whether its value equals S or differs from S. Justify your conclusion without performing numerical computation beyond the first few terms.
(c) Repeat part (b) for the series S2.
(d) Based on your work in parts (a)–(c), explain in general terms why some rearrangements of S must preserve the sum while others may alter it or even fail to converge.