Question

If `a_n` converges and `a_n >b_n` for all n, does `b_n` converge?

Answer 1

See explanation below

Explanation:

We assume that we are talking about sequences (although, for infinite series, the reasoning is the quite similar)

Assume that `{a_n}` converges. This is the same `lim_(ntooo)a_n=L` with `L` a finite number

For all `{b_n}` we know that `b_n < a_n`

If `{b_n}` is a growing sequence, then all terms are under the value `L` and `{b_n}` converges perhaps to `L`

If `{b_n}` is a decreasing or alternate sequence, we can´t assume the convergence. Think in this example

`a_n=3+1/n` and `b_n=(-1)^n`

In this case `b_n < a_n` for all `n`, but `lim_(ntooo)a_n=3` and
`lim_(ntooo)b_n` doesn`t exist