Thread: a few true or false questions about sequences/series and convergence

hi y'all, i'm having some real trouble with these T or F problems. I've tried to do them myself, but i have no idea if my answers are right. I would appreciate if you could tell me what you think the answers are so i can compare them to my own. Thanks!


a.) a sequence {a[subscript]n} only converges if
lim[subscript]n->infiniti a[subscript]n = 0


b.) if lim[subscraipt]n->infiniti (a[subscript]n / b[subscript]n) = 1 , where
a[subscript]n and b[subscript]n are positive for all n , and
[Epsilon]infinity,n=1 of b[subscript]n converges,
then [Epsilon]infinity,n=1 of a[subscript]n also converges.


c.) if lim[subscript]n->infiniti a[subscript]n = 0 ,
then [Epsilon]infinity,n=0 of a[subscript]n converges.


d.) If a series is absolutly convergent, then it is convergent.


e.) if a series is conditionaly convergent, then the terms of a series can be arranged so that the sum of ther series is pi.

and your answers are ?

ok, but like i said, i'm really uncertain about these, except problems a and d, which i at least have some idea of what i'm doing(ish).

a.) true

b.) true

c.) true

d.) true

e.) true

a) is false. A sequence will converge if it has any limit . The sequence will diverge if the limit does not exist or the limit tends to .