Converges?

[fading bobcat]

In blue what I think the solution is.
In black what GPT says.

[fading bobcat]

Both GPT and me get the solution right, I think

Even that, am I making a mistake?
What theorem would you use?

[knotty crow]

calc-2?

[thick crypt]

1/(1 + a^n) < 1/a^n So, sqrt(a^n)/(1 + a^n) < sqrt(a^n)/a^n = 1/sqrt(a^n) = 1/sqrt(a)^n So the sum converges if sqrt(a) < 1 which is the same as saying a < 1. Hence, your final condition is 0 <= a < 1

Only theorems needed are direct comparison test and geometric serie convergence