Question

I'm studying a back test for my Mathematical Analysis course, but I'm not sure how to answer one of the questions. Give an example of a sequence {yn} such that lim (n->Infinity) |yn-y(n+1)|=0 but is not a Cauchy Sequence.

Answer 1

i don't have a ready answer, but i have an idea

Answer 2

oh good Q

Answer 3

the difference between this and a cauchy sequence is that this says adjacent elements of the sequence converge to zero, not arbitrary ones

Answer 4

got it

Answer 5

try \(a_n\) as the nth partial sum of the harmonic series

Answer 6

i think that is right, and it generalizes to
\[\lim|a_{n+r}-a_n|=0\]

Answer 7

for fixed \(r\) of course.

Answer 8

@Aylin is the answer clear?

Answer 9

I was just about to come back and ask if y{1}=1 and y{n+1}=y{n}+1/n would work.

Answer 10

I think so.

Answer 11

point is the harmonic series diverges, so you can never find an \(N\) such that if \(m,n>N\) you would have
\[\sum_1^n\frac{1}{k}-\sum_1^m\frac{1}{k}<\epsilon\] in fact this difference can be made larger than and larger. but
\[\sum_1^n\frac{1}{k}-\sum_1^{n+1}\frac{1}{k}=\frac{1}{n+1}\]

Answer 12

Thank you.

Answer 13

yw