Real Analysis: Proving this series is divergent

[hoary cosmos]

I've tried proving the contrapositive and using the definition of convergence/divergence to no avail. I've tried the limit comparison test as well. Is there any other way I could approach this problem?

[candid gyro]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[sage mortar]

Hmmm well the way I’d do it is study all the possible cases:
Case1: the sequence an diverges than the series an/(1+an) diverges because it doesn’t have a null limit
Case 2: an converges to a non 0 limit , so an/1+an also converges to an non 0 limit thus the series diverges

Case 3: an converges to 0 than an/(an/(1+an)=1+an

Knowing an converges to 0 we have an~(an((1+an)) but the series an diverges thus the series an/(1+an) diverges as well

[sage mortar]

I think this is a way to do it but not sure

[hoary cosmos]

I actually managed to prove it now but using a different method. Thanks for replying!

+close