## 4.1. Series and Convergence

### Examples 4.1.7(b):

Take a look at the following inequality:for2^{n}+ j < 2^{n+1}

*j = 1, 3, ..., 2*(which makes for

^{n}-1*2*terms). Therefore:

^{n-1}But now we can rearrange the terms of the alternating harmonic series as follows:(2^{n}+ 1)^{-1}+ (2^{n}+ 3)^{-1}+ ... + (2^{n+1}- 1)^{-1}> 2^{n-1}/ 2^{n+1}=^{1}/_{4}

Since each term in square brackets is greater than1 -^{1}/_{2}+^{1}/_{3}-^{1}/_{4}+

[ (^{1}/_{5}+^{1}/_{7}) -^{1}/_{6}] +

[ (^{1}/_{9}+^{1}/_{11}+^{1}/_{13}+^{1}/_{15}) -^{1}/_{8}] + ...

+ [ ( (2^{n}+1)^{-1}+ (2^{n}+3)^{-1}+ ... + (2^{n+1}-1)^{-1}) -^{1}/_{2n+2}] + ...

*the rearranged series must diverge to positive infinity.*

^{1}/_{4}-^{1}/_{6}