Suggested languages for you:

Americas

Europe

Problem 1135

Establish the convergence or divergence of the series: $[1 /(1+\sqrt{1})]+[1 /(1+\sqrt{2})]+[1 /(1+\sqrt{3})]+[1 /(1+\sqrt{4})]+\ldots$

Expert verified

The series \(\sum_{n=1}^{\infty} \frac{1}{1 + \sqrt{n}}\) diverges, as demonstrated by the Comparison Test with the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\), which also diverges. The inequality \(\frac{1}{1 + \sqrt{n}} \geq \frac{1}{n}\) holds for all \(n \geq 1\), showing that the terms of the given series are larger than those of the harmonic series.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 36

Determine the general term of the sequence: $$ 1 / 2,1 / 12,1 / 30,1 / 56,1 / 90, \ldots $$

Chapter 36

Establish the convergence or divergence of the series: $\sin \pi / 2+1 / 4 \sin \pi / 4+1 / 9 \sin \pi / 6+1 / 16 \sin \pi / 8+\ldots .$

Chapter 36

Determine the general term of the sequence: $$ 1 / 5^{3}, 3 / 5^{5}, 5 / 5^{7}, 7 / 5^{9}, 9 / 5^{11} $$

Chapter 36

Test the series: $\left[1-3^{2} / 2^{2}\right]+\left[3^{4} /\left(2^{2} \cdot 4^{2}\right)\right]-\left[3^{6} /\left(2^{2} \cdot 4^{2} \cdot 6^{2}\right)\right]+\ldots \ldots \ldots$ by means of the ratio test. If this test fails, use another test.

Chapter 36

Test the series: \(1+2 ! / 2^{2}+3 ! / 3^{3}+4 ! / 4^{4}+\ldots \ldots\) by means of the ratio test. If this test fails, use another test.

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner