# Question on series

• December 1st 2009, 12:16 PM
katielaw
Question on series
Given that x_i - y_i goes to zero as i goes to infinity and that the ∑ x_i converges, show that ∑ y_i may diverge.

I don't understand this question. Am I supposed to prove this using ∑ y_i as some arbitrary sequence, or can I come up with some example with real numbers?
• December 1st 2009, 12:42 PM
Krizalid
let $x_i=\frac1{i^2},$ $y_i=\frac1i,$ clearly the difference goes to zero, but $\sum y_i$ diverges.
• December 1st 2009, 12:44 PM
Enrique2
You have simply to put a concrete example such that this situation happens. take $y_i=\frac{1}{i}$, and $x_i$ any convergente sequence.