Question

Absolute and Conditional ConvergenceWhich of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]

Accepted Answer

Identify the given series: $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{1 + \sqrt{n}} . $$This is an alternating series because of the factor $$ (-1)^n $$, which alternates the sign of each term. To check for absolute convergence, consider the series of absolute values: $$ \sum_{n=1}^{\infty} \left| \frac{(-1)^n}{1 + \sqrt{n}} ight| = \sum_{n=1}^{\infty} \frac{1}{1 + \sqrt{n}} . $$Analyze whether this series converges or diverges. Compare the absolute value series to a known benchmark series. Since $$ \sqrt{n} $$ grows slower than $$ n $$, the terms $$ \frac{1}{1 + \sqrt{n}} $$ behave similarly to $$ \frac{1}{\sqrt{n}} $$ for large $$ n . $$Recall that $$ \sum \frac{1}{n^p} $$ converges if and only if $$ p \u003e 1 . $$Since $$ \sum \frac{1}{\sqrt{n}} = \sum \frac{1}{n^{1/2}} $$ diverges (because $$ 1/2 \u003c 1 $$), the absolute value series diverges. Therefore, the original series does not converge absolutely. Next, apply the Alternating Series Test to the original series: check if the terms $$ b_n = \frac{1}{1 + \sqrt{n}} $$ decrease monotonically to zero. Since $$ b_n  is $$positive, decreasing, and $$ \lim_{n 	o \infty} b_n = 0 $$, the series converges conditionally.

Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]

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Concetti chiave

Absolute Convergence

Conditional Convergence

Alternating Series Test