https://file.aiquickdraw.com/mathbot/file/1740640192115jujvuasm.jpg

https://file.aiquickdraw.com/mathbot/file/1740640192116gk0f9lr4.jpg

Let's work through the problem in the first image you've provided, using the comparison test similar to the example in the second image.

The series you've written is:

\[
\sum_{n=1}^{\infty} \frac{1 + \sin(n)}{n \cdot 10^n}
\]

### Step 1: Define \( a_n \) and \( b_n \)
We define the series terms for comparison:
- \( a_n = \frac{1 + \sin(n)}{n \cdot 10^n} \)
- \( b_n = \frac{1}{n \cdot 10^n} \)

### Step 2: Find the Limit of the Ratio \( \frac{a_n}{b_n} \)
Next, we calculate the limit of the ratio:

\[
\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{1 + \sin(n)}{n \cdot 10^n}}{\frac{1}{n \cdot 10^n}} = \lim_{n \to \infty} (1 + \sin(n))
\]

Since \( \sin(n) \) oscillates between -1 and 1, the expression \( 1 + \sin(n) \) will always be between 0 and 2. Therefore, the limit of the ratio will be a finite value:

\[
\lim_{n \to \infty} \frac{a_n}{b_n} = \text{bounded value between 0 and 2}.
\]

### Step 3: Use the Limit Comparison Test
The series \( \sum_{n=1}^{\infty} \frac{1}{n \cdot 10^n} \) is a convergent series because it is of the form \( \sum_{n=1}^{\infty} \frac{1}{n r^n} \), which converges for any \( r > 1 \) (in this case, \( r = 10 \)).

Since \( \lim_{n \to \infty} \frac{a_n}{b_n} \) is a finite positive value, we can conclude that the original series behaves similarly to the comparison series. Therefore, by the limit comparison test, the series:

\[
\sum_{n=1}^{\infty} \frac{1 + \sin(n)}{n \cdot 10^n}
\]

**converges.**

---

Would you like more details on this solution, or do you have any questions about the steps? Here are some related questions to help expand on this topic:

1. How does the comparison test work with divergent series?
2. Can you explain how the Limit Comparison Test helps to determine convergence in a series?
3. What are some other tests to determine convergence or divergence of infinite series?
4. How does oscillating behavior of functions like \( \sin(n) \) affect the convergence of a series?
5. What is the difference between absolute and conditional convergence in infinite series?

**Tip:** When using the limit comparison test, ensure that the comparison series is known to converge or diverge, and the limit of their ratio exists as a finite positive number.

Learn how to determine the convergence of the series \sum_{n=1}^{\infty} \frac{1 + \sin(n)}{n \cdot 10^n} using the Limit Comparison Test. This solution explains the steps involved, including defining terms, calculating limits, and applying the comparison test for series convergence. Suitable for college-level students studying series and convergence tests.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation