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Considertheseriesk=1(-1)k+1akandk=1(-1)kakwhere(i){ak}isasequenceofpositivenumber(ii)thesequence{ak}isstrictlydecreasing(iii)limkak=0

Find an example of a divergent series of the form

(a) that satisfies conditions (i) and (iii), but not condition (ii);

(b) that satisfies conditions (i) and (ii), but not condition (iii).

Short Answer

Expert verified

(i)(i)Theseriesk=1(-1)k+1aksatisfyingthegivenconditionisk=1(-1)k+12k.(ii),Theseriesk=1(-1)k+1aksatisfyingthegivenconditionisk=1(-1)k+1k2k-1.

Step by step solution

01

Step 1. Given

Considertheseriesk=1(-1)k+1akandk=1(-1)kakwhere(i){ak}isasequenceofpositivenumber(ii)thesequence{ak}isstrictlydecreasing(iii)limkak=0

02

Step 2. Hypothesis of alternating series

Itstatesthatif{ak}isasequenceofpositivenumberwithak+!<akforeveryk1andlimkak=0thenalternatingseriesk=1(-1)k+1akandk=1(-1)kakbothconvergeisak+1<ak

03

Part(a) Step 3. Explanation

Considertheseriesk=1(-1)k+1ak=k=1(-1)k+12kThesequenceak=12kisapositivetermsThevalueoflimkak=limk1k=0Buttheseriesk=1(-1)k+1ak=k=1(-1)k+12kisanincreasingseries.Thus,theseriesk=1(-1)k+1aksatisfyingthegivenconditionisk=1(-1)k+12k.

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Part(b) Step 4. Explanation

Considertheseriesk=1(-1)k+1ak=k=1(-1)k+1k2k-1Thesequenceak=k2k-1isapositivetermsThevalueoflimkak=limkk2k-1=12Thus,theseriesk=1(-1)k+1aksatisfyingthegivenconditionisk=1(-1)k+1k2k-1.

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