Determine if the sequence converges or diverges

Question

an=(1+(1/n³))^n

Answer 1

This one is built to look like a famous limit, so the first thing to do is notice that it is not quite that limit. You have probably seen:

lim (n→∞) (1 + 1/n)ⁿ = e

and this problem looks close enough that it is tempting to answer e. But look at the exponent inside compared to the exponent outside. Here we have 1/n³ inside and only n outside, and that mismatch changes the answer completely.

The reliable way to handle anything with n in the exponent is to take the natural log first, find that limit, and then undo the log at the end. So let:

ln a_n = n · ln(1 + 1/n³)

Now, when u is small, ln(1 + u) behaves like u. Since 1/n³ is getting very small, we get:

ln a_n ≈ n · (1/n³) = 1/n²

And 1/n² goes to 0. So ln a_n → 0, which means:

a_n → e⁰ = 1

If you notice, this is exactly why the famous limit gives e and this one does not. In (1 + 1/n)ⁿ you get n · (1/n) = 1, so the log limit is 1 and the answer is e¹. Here the n outside cannot keep up with the n³ inside, the log limit collapses to 0, and you land on e⁰ instead. The whole problem comes down to comparing those two powers.

You can watch it happen: a_n is about 1.0100 at n = 10, then 1.000100 at n = 100, then 1.0000010 at n = 1000. It is squeezing down onto 1, not climbing toward 2.718.

So, our final answer is that the sequence converges, and it converges to 1.