https://r2.erweima.ai/mathbot/image/1725442672307wts4uwqp.jpg

The problem in the image asks to show that the series 

\[
\sum_{n=0}^{\infty} \frac{1}{\sqrt{n+1}}
\]

converges. The hint suggests determining the convergence by considering the series of absolute values.

### Solution:

To determine the convergence of the series

\[
\sum_{n=0}^{\infty} \frac{1}{\sqrt{n+1}},
\]

we can apply the **Comparison Test** or **Integral Test** since the series terms are positive.

#### **Step 1: Check the series using the Comparison Test**

Consider the function \( f(n) = \frac{1}{\sqrt{n+1}} \). Notice that as \( n \) becomes large, \( \frac{1}{\sqrt{n+1}} \) behaves similarly to \( \frac{1}{\sqrt{n}} \), because:

\[
\frac{1}{\sqrt{n+1}} \sim \frac{1}{\sqrt{n}} \text{ as } n \rightarrow \infty.
\]

Now, compare this with the known p-series \( \sum \frac{1}{n^p} \):

- If \( p > 1 \), the series converges.
- If \( p \leq 1 \), the series diverges.

Here, for \( \frac{1}{\sqrt{n}} \), \( p = \frac{1}{2} \). Since \( p \leq 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) diverges.

However, this suggests that \( \sum_{n=0}^{\infty} \frac{1}{\sqrt{n+1}} \) may also diverge, but we need to examine this more carefully.

#### **Step 2: Check the series using the Integral Test**

Another approach is to apply the Integral Test. Consider the integral

\[
\int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx.
\]

Calculate the integral:

\[
\int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C.
\]

Evaluate the improper integral:

\[
\int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx = \lim_{b \to \infty} \left[ 2\sqrt{x} \right]_{1}^{b} = \lim_{b \to \infty} \left( 2\sqrt{b} - 2 \right).
\]

As \( b \) approaches infinity, \( 2\sqrt{b} \) also approaches infinity. Thus, the integral diverges, indicating that the original series diverges as well.

### Conclusion:

The series \( \sum_{n=0}^{\infty} \frac{1}{\sqrt{n+1}} \) **diverges**.

---

Do you want further details or have any questions?

Here are some related questions:
1. What is the p-series test and how is it applied to determine convergence?
2. How does the Comparison Test work for series convergence?
3. Can the ratio test be applied to determine the convergence of this series?
4. What is the Integral Test, and why is it used to determine series convergence?
5. How do series of the form \( \sum \frac{1}{n^p} \) differ in convergence based on the value of \( p \)?

**Tip:** For series involving \( n^{-\alpha} \), consider whether \( \alpha \) is greater than 1 for convergence using the p-series test.

Sei φ ∈ R. Zeigen Sie, die Reihe Σ(n=0 bis ∞) 1/√(n+1) konvergiert.

Learn how to prove the convergence of the series Σ(n=0 bis ∞) 1/√(n+1) using the Comparison Test and the Integral Test. This detailed step-by-step solution covers key concepts in series convergence and is suitable for undergraduate mathematics students.

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