Squeeze theorem

Question

Given the sequence (3ⁿ + 1) ^1/n, what would the upper bound be if the squeeze theorem was used to prove that the sequence converges for n approaching infinity?

Answer 1

isn't 1+3ⁿ <4ⁿ for n>1

so the limit is less than 4

Answer 2

Dayv is, of course, correct.

And since 3ⁿ<3ⁿ+1, you have 3<(3ⁿ+1)^1/n<4.

This tells you that the sequence converges, but does not give you the limit.

The so-called Squeeze Theorem is usually used to prove the limit...which means better estimates need to used to apply that theorem.

Right away, I don't see a way to get a better estimate for the upper bound.

OK, try this.

x_n = 3(1+3^-n)^1/n

The first three terms of the binomial are 3[1 + (1/n)3^-n - n(n-1)3^--2n/2n²...

This is an alternating sign series so the the error does not exceed in absolute value the first omitted term.

This gives you the upper and lower bound you need for the so-called Squeeze Theorem and shows that the limit is, in fact, 3.