Use the infinite alternating p-series sigma^infinity_n = 1(-1)^n/Squareroot n to

Question

Use the infinite alternating p-series sigma^infinity_n = 1(-1)^n/Squareroot n to answer questions 5 through 7 List the first five terms of the sequence generated by a_n. Show that the alternating series, sigma^infinity_n = 1 is conditionally convergent. Verify each of the three conditions for an alternating series to be convergent. Show all necessary calculations to support your conclusion. Is the alternating series absolutely convergent? (Would it still be convergent if the alternating sign, (-1)^n, were removed?) Identify the p-series term used for comparison and state whether it is convergent or divergent. Show all necessary calculations to support your conclusion.

Explanation / Answer

given series [n=1 to ](-1)n/n

5) five terms are (-1)1/1,(-1)2/2,(-1)3/3,(-1)4/4,(-1)5/5

five terms are -1,1/2,-1/3,1/2,-1/5

6,7)[n=1 to ]|(-1)n/n|

=[n=1 to ] 1/n

=[n=1 to ] 1/n1/2 is divergent by p series test (p=1/2 <1)

bn=1/n

limn-> bn

=limn-> 1/n

=0

bn=1/n is a decreasing sequence . so [n=1 to ](-1)n/n is convergent by alternating series test.

[n=1 to ](-1)n/n is convergent ,[n=1 to ] 1/n is divergent

therefore [n=1 to ](-1)n/n is conditionally convergent

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.