This is a problem on convergence that was confusing me, so any help would be appreciated!
*Note that when I say Sigma below, I mean the sigma symbol (I'm not sure how to input it so I just wrote the word)
Suppose Sigma starting with j = 0 and ending at infinity of b_j converges and a_j > b_j for all j>16. What can we conclude about Sigma starting with j = 0 and ending at infinity for a_j?
(Meaning, does it converge, diverge, etc?)
Now if b_j > a_j, then we'd be in business
Consider the sequence of partial sums,
And so on,
Is the sequence of partial sums.
This sequence has no limit.
There are two ways to show it has no limit.
Either violate the epsilon definition.
But here is an easier way.
Assume that is has a limit (real or infinite).
Then, all subsequences must have the same limit (theorem 11.2).
a_even = 0,0,0,0,...
b_odd = -1,-1,-1,...
Are two subsequences that converge to two different values, i.e. -1 and 0 irrespectively.
Thus, there is no limit.