Analysis of the Convergence or Divergence of the Series -11n/√n

When analyzing the convergence or divergence of the series (sum_{n1}^{infty} frac{-1^{1n}}{sqrt{n}}), we need to understand the behavior of its terms as (n) approaches infinity.

Rewriting the Series

The given series can be rewritten as:

[frac{-1^{1n}}{sqrt{n}} frac{(-1)^{n-1}}{sqrt{n}} frac{(-1)^{n-1}}{n^{1/2}}]

This expression represents an alternating series, which can be analyzed using the Alternating Series Test.

Applying the Alternating Series Test

The Alternating Series Test states that an alternating series of the form (sum (-1)^n a_n) converges if the following conditions are met:

(a_n) is positive for all (ngeq 1). (a_n) is decreasing for all sufficiently large (n). The limit [lim_{n to infty} a_n 0].

For our series, we have:

(a_n frac{1}{sqrt{n}}).

Checking the Conditions

Positivity

To check positivity, observe that for all (n geq 1):

[a_n frac{1}{sqrt{n}} geq 0]

Decreasing

Next, we need to check if the sequence (a_n) is decreasing. Compute (a_{n 1}) and check if:

[a_{n 1} frac{1}{sqrt{n 1}} leq frac{1}{sqrt{n}} a_n]

This inequality holds true for all (n geq 1), which confirms that the sequence is decreasing.

Limit

Finally, compute the limit:

[lim_{n to infty} a_n lim_{n to infty} frac{1}{sqrt{n}} 0]

Since all three conditions of the Alternating Series Test are satisfied, we can conclude that the series:

(sum_{n1}^{infty} frac{(-1)^{n-1}}{sqrt{n}})

converges.

Conclusion

The series (sum_{n1}^{infty} frac{(-1)^{n-1}}{sqrt{n}}) converges, as all the conditions of the alternating series test are met:

The terms are decreasing. The limit of the terms is zero. The terms themselves are positive.

This analysis demonstrates that the series (sum_{n1}^{infty} frac{-1^{1n}}{sqrt{n}}) indeed converges.