Do the series that converge to a positive number form a subs

Do the series that converge to a positive number form a subspace of the vector space of convergent series\'? How about the series that converge absolutely?

Solution

Answer to first question: No

eg. 1,1,1,1,1,1,......................... is a series converges to 1

If it were to form a vector space then multiplying it by a scalar would give us another vector in the vector space.

Multiplying by -1 gives

-1,-1,-1,................ is a series that converges to -1

Hence not a vector space

Answer to second part . Yes

1. Constant series with all terms 0 converges absolutely ie 0 belongs to this set

2. Let, an ,bn belong to this set

Hence, |an|,|bn| converge absolutely

|an+bn|<=|an|+|bn|

Hence by comparison test

an+bn converges absolutely.

HEnce it forms a vector space.