Find two series a_n and b_n such that (a_n + b_n) are convergent but a_n and b_n a...

[biogirl]

I need to find two series, a_n and b_n, which are both divergent individually, but the series (a_n + b_n) is convergent.

I tried a_n = 1/n, and b_n = 1/n^1/2. Both of these are divergent p-series, and I found that (a_n + b_n) = (1 + n^1/2)/n. The limit of this (a_n + b_n) is the part I'm having trouble with to find out if it's convergent or divergent. I found the ratio test difficult to use, and evaluating the limit directly resulted in 0. But I wonder, is the right way to do this to compare it to a p-series, with p = 1/2, and therefore (a_n + b_n) is also divergent?

Thanks.

 

[stapel]

No need to get complicated. You know that the series a_n = n diverges. You also know that the series b_n = -n diverges. What about the sum?

 

[biogirl]

Ah, lovely! Thank you for the suggestion. Now I think I can approach the other questions a bit more easily too. Greatly appreciated.