https://file.aiquickdraw.com/mathbot/file/1725356147910s02g94p2.jpg

1) Найти сумму n первых членов ряда Sn. 2) доказать сходимость ряда пользуясь определением сходимости и 3)найти сумму ряда S

The image contains a mathematical problem involving infinite series. Here’s a breakdown of the tasks:

1. **Find the sum of the first \( n \) terms of the series \( S_n \)**.
2. **Prove the convergence of the series using the definition of convergence**.
3. **Find the sum of the series \( S \)**.

To address each of these points, I'll first transcribe and analyze the specific series from the image, then solve each part accordingly.

---

### Step 1: Transcription of the Series

The series are given as follows:

1. \[
S_n = 1 + \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^{n-1}}
\]
2. \[
S_n = 1 - \frac{1}{2} + \frac{1}{4} - \cdots + (-1)^{n-1} \frac{1}{2^{n-1}}
\]
3. \[
S_n = 1 + \frac{1}{3} + \frac{1}{9} + \cdots + \frac{1}{3^{n-1}}
\]

---

### Step 2: Solving Each Part

**1. Finding the sum of the first \( n \) terms \( S_n \):**

1. For the first series:
   - This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{2} \).
   - The sum of the first \( n \) terms is:
     \[
     S_n = \frac{1 \cdot (1 - \left(\frac{1}{2}\right)^n)}{1 - \frac{1}{2}} = 2 \left(1 - \frac{1}{2^n}\right)
     \]

2. For the second series:
   - This is an alternating geometric series with first term \( a = 1 \) and common ratio \( r = -\frac{1}{2} \).
   - The sum of the first \( n \) terms is:
     \[
     S_n = \frac{1 \cdot (1 - (-\frac{1}{2})^n)}{1 - (-\frac{1}{2})} = \frac{2}{3} \left(1 - (-\frac{1}{2})^n\right)
     \]

3. For the third series:
   - This is also a geometric series with first term \( a = 1 \) and common ratio \( r = \frac{1}{3} \).
   - The sum of the first \( n \) terms is:
     \[
     S_n = \frac{1 \cdot (1 - \left(\frac{1}{3}\right)^n)}{1 - \frac{1}{3}} = \frac{3}{2} \left(1 - \frac{1}{3^n}\right)
     \]

**2. Proving the Convergence:**

To prove convergence, we examine the limit of the sum \( S_n \) as \( n \) approaches infinity.

- For the first series:
  \[
  \lim_{n \to \infty} S_n = \lim_{n \to \infty} 2 \left(1 - \frac{1}{2^n}\right) = 2 \times 1 = 2
  \]
  The series converges to 2.

- For the second series:
  \[
  \lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{2}{3} \left(1 - (-\frac{1}{2})^n\right) = \frac{2}{3} \times 1 = \frac{2}{3}
  \]
  The series converges to \(\frac{2}{3}\).

- For the third series:
  \[
  \lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{3}{2} \left(1 - \frac{1}{3^n}\right) = \frac{3}{2} \times 1 = \frac{3}{2}
  \]
  The series converges to \(\frac{3}{2}\).

**3. Finding the sum of the series \( S \):**

Given the convergence proof, the sums of the series are:

1. **Sum of the first series**: \( S = 2 \)
2. **Sum of the second series**: \( S = \frac{2}{3} \)
3. **Sum of the third series**: \( S = \frac{3}{2} \)

---

Would you like a detailed explanation of any part of this process? Here are five related questions to deepen your understanding:

1. How do you determine the convergence of a geometric series?
2. Why does the alternating series converge more slowly?
3. What is the significance of the common ratio in determining the sum of a series?
4. Can the method used here be applied to non-geometric series?
5. How would the convergence proof differ for a divergent series?

**Tip:** Always check the common ratio in a geometric series; if the absolute value of the common ratio is less than 1, the series will converge.

Learn how to find the sum of the first n terms of a series, prove the convergence of a series using convergence criteria, and determine the sum of the entire series. This problem involves geometric and alternating series, with step-by-step solutions and explanations. Suitable for higher education level studies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation