Real Analysis: proving a sequence converges and finding its limit.

[TeenieBopper]

Homework Statement

Suppose r>1. Prove the sequence \sqrt[n]{1 + r^{n}} converges and find its limit.

Homework Equations

The Attempt at a Solution

It's obvious that the sequence converges to r, so I know where I need to end up. My first instinct is to use the squeeze theorem. It's obvious that \sqrt[n]{r^{n}}<\sqrt[n]{1 + r^{n}}. However, I'm having difficulty finding a sequence that's greater than \sqrt[n]{1 + r^{n}} but also converges to r.

 

[SammyS]

Homework Statement

Suppose r>1. Prove the sequence \sqrt[n]{1 + r^{n}} converges and find its limit.

Homework Equations

The Attempt at a Solution

It's obvious that the sequence converges to r, so I know where I need to end up. My first instinct is to use the squeeze theorem. It's obvious that \sqrt[n]{r^{n}}<\sqrt[n]{1 + r^{n}}. However, I'm having difficulty finding a sequence that's greater than \sqrt[n]{1 + r^{n}} but also converges to r.

Try comparing to \displaystyle \sqrt[n]{2\,r^n}\ .