Homework SourseIf sigma a n converges and b n converges then sigma a n b n
If sigma a_ n converges and (b_ n) converges, then sigma a_ n b_ n converges.
Solution
Answer :
Let ( bn ) converges then ( bn ) is monotonic and bounded.
Suppose bn b R . Since ( bn ) is monotonic . we either bn+1 bn for all n ( non - decreasing ) or bn+1 bn for all n ( non - increasing ) .
Assume, without loss of generality, that ( bn ) is monotonic non- decreasing, otherwise consider - bn
if ( bn ) is non - decreasing then bn b for all n.
Consider cn = b - bn we have that an converges, the partial sum An of an form a bounded sequence;
C0 C1 C2 C3 . . . Cn 0
Thus by known fact , that anbn converges.
