   Chapter 11.2, Problem 86E

Chapter
Section
Textbook Problem

If ∑ a n and ∑ b n are both divergent, is ∑ ( a n + b n ) necessarily divergent?

To determine

To check:

If an and bn are both divergent then (an+bn) is necessarily divergent or not.

Explanation

1) Concept:

Choose examples of divergent series for both an and bn then check (an+bn) is divergent or not.

2) Given:

an and bn are both divergent

3) Calculation:

For example:

an=1nand bn=(-1 n)

Consider,

an=1n

For the particular series n=11n, consider the partial sum of s2,s4,,s8,s16,s32,

s2=1+12

s4=1+12+13+14

s8=1+12+13+14+15+16+17+18

s8>1+12+14+14+18+18+18+18

=1+12+12+12=1+32

s8>1+32

s16=1+12+13+14+15+16+17+18+19+110+111++116

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