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.
See the step by step solution

## Step 1: Write the series and the harmonic series for comparison

We have the given series: $\sum_{n=1}^{\infty} \frac{1}{1 + \sqrt{n}}$ And the harmonic series: $\sum_{n=1}^{\infty} \frac{1}{n}$

## Step 2: Compare the terms of the series

In order to compare the terms of the given series and the harmonic series, we will analyze the inequality: $\frac{1}{1 + \sqrt{n}} \geq \frac{1}{n}$ For all $$n \geq 1$$, it is easy to see that $$\sqrt{n} \geq 1$$, which implies that $$1 + \sqrt{n} \geq 2$$. Therefore, $$n \geq 1 + \sqrt{n}$$. Consequently, we have: $\frac{1}{n} \leq \frac{1}{1 + \sqrt{n}}$

## Step 3: Apply the Comparison Test

Since the terms of our series are larger than the terms of the harmonic series, which is known to diverge, then by the Comparison Test, our series also diverges. Thus, the series: $\sum_{n=1}^{\infty} \frac{1}{1 + \sqrt{n}}$ diverges.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

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