   Chapter 11.5, Problem 35E

Chapter
Section
Textbook Problem

Show that the series ∑ ( − 1 ) n − 1 b n , where b n = 1 / n if n is odd and b n = 1 / n 2 if n is even, is divergent. Why does the Alternating Series Test not apply?

To determine

To show:

(i) -1n-1bn is divergent.

(ii) Give reasons as to why the Alternating Series Test doesn’t apply.

Explanation

1) Concept:

i) Use theorem 11.2.8 ii to show -1n-1bn is divergent.

ii) Alternating Series Test:

If the alternating series

n=1-1nbn,  bn>0

satisfies,

i) bn+1<bn for all  n

ii) limnbn=0

then the series is convergent otherwise the series is divergent.

2) Theorem 11.2.8 ii:

If an and bn are convergent series, then, so are the series can (where c is a constant),

an+bn and an-bn, and

(i) n=1can=cn=1an

(ii) n=1an+bn=n=1an+n=1bn

(iii) n=1an-bn=n=1an-n=1bn

3) Given:

-1n-1bn

where bn=1n if n is odd

and bn=1n2 if n is even.

4) Calculation:

(i)

If n is even

b2n=12n2

By comparison with the p-series test for p=2, the series b2n converges

Suppose on contrary,

-1n-1bn converges

b2n converges and -1n-1bn converges

By theorem 11

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