Question

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=1+(-1)^{n}\)

Short answer

The sequence \(a_{n}=1+(-1)^{n}\) does not converge. It diverges because it does not approach a limit as \(n\) goes to infinity.

Step-by-step solution

Determine the behaviour of the sequence

For the sequence \(a_{n}=1+(-1)^{n}\), when \(n\) is an even number, the -1 raised to an even power is 1, so the sequence term becomes \(a_{n} = 1+1 = 2\). When \(n\) is an odd number, -1 raised to an odd power is -1, so the term becomes \(a_{n} = 1-1 = 0\).

Analyse the limit of the sequence

Since the sequence fluctuates between 0 and 2, depending on whether \(n\) is odd or even, it does not approach a specific number as \(n\) goes to infinity. Therefore, the limit of the sequence does not exist.

Conclusion about the convergence or divergence

As the sequence does not approach any limit as \(n\) goes to infinity, we can conclude that the sequence diverges.