Alllylic bromination of 1-butene My book asks me to consider reacting 1-butene with NBS and light. 

I know that NBS and light leads to radical formation, and this is an allylic/benzylic bromination problem. 

However, my book only provides one product. 

Aren't both products expected? Actually, I would expect one product in excess (the book actually listed my minor product as the sole product). There are at least two resonance contributors, and the more major one has the more substituted radical.
Yes you are correct, at least two products are formed (ref. 1, ref. 2).  Unfortunately, neither of these references provide the actual product distribution between the isomers.

I say that "at least two products are formed" because depending on your reaction conditions 3 products might be expected.  NBS brominations can be carried out over a range of temperatures, from cold to somewhat above room temperature.  The activation energy required to rotate about an allylic bond is only ~15-16 kcal/m (see here).  Therefore, your 1-bromo isomer could exist as a cis-trans mixture, unless the reaction is performed at very cold conditions.

In the analogous case of NBS bromination of 1-octene, a mixture of 3-bromo-1-octene (~18%) and 1-bromo-2-octene (~82%) - both cis and trans are produced. 

$\ce{RCH2CH=CH2  +  NBS \rightarrow RCHBrCH=CH2  +  RCH=CHCH2Br}$ (cis &amp; trans)