Find the first five partial sums of the given series and determine whether the series appears to be
convergent or divergent. If it is convergent, find its approximate sum.

The first partial sum is .
(Type an integer or decimal rounded to four decimal places as needed.)

Ask by Bowers Hampton. in the United States

Jan 12,2025

Question

Find the first five partial sums of the given series and determine whether the series appears to be
convergent or divergent. If it is convergent, find its approximate sum.

The first partial sum is .
(Type an integer or decimal rounded to four decimal places as needed.)

Ask by Bowers Hampton. in the United States

Jan 12,2025

UpStudy AI · Accepted Answer

To find the first five partial sums of the series $$ \sum_{n=1}^{\infty} \frac{3}{(n+1)(n+2)} $$, let's start by computing each partial sum individually.

### Partial Sums

1. **First Partial Sum ($$ S_1 $$)**
   $$
   S_1 = \frac{3}{(1+1)(1+2)} = \frac{3}{2 \times 3} = \frac{3}{6} = 0.5
   $$
   **$$ S_1 = 0.5 $$**

2. **Second Partial Sum ($$ S_2 $$)**
   $$
   S_2 = S_1 + \frac{3}{(2+1)(2+2)} = 0.5 + \frac{3}{3 \times 4} = 0.5 + \frac{3}{12} = 0.5 + 0.25 = 0.75
   $$
   **$$ S_2 = 0.75 $$**

3. **Third Partial Sum ($$ S_3 $$)**
   $$
   S_3 = S_2 + \frac{3}{(3+1)(3+2)} = 0.75 + \frac{3}{4 \times 5} = 0.75 + \frac{3}{20} = 0.75 + 0.15 = 0.9
   $$
   **$$ S_3 = 0.9 $$**

4. **Fourth Partial Sum ($$ S_4 $$)**
   $$
   S_4 = S_3 + \frac{3}{(4+1)(4+2)} = 0.9 + \frac{3}{5 \times 6} = 0.9 + \frac{3}{30} = 0.9 + 0.1 = 1.0
   $$
   **$$ S_4 = 1.0 $$**

5. **Fifth Partial Sum ($$ S_5 $$)**
   $$
   S_5 = S_4 + \frac{3}{(5+1)(5+2)} = 1.0 + \frac{3}{6 \times 7} = 1.0 + \frac{3}{42} \approx 1.0 + 0.0714 = 1.0714
   $$
   **$$ S_5 \approx 1.0714 $$**

### Convergence Analysis

To determine whether the series is convergent or divergent, let's consider the general approach:

1. **Expressing as a Telescoping Series:**
   $$
   \frac{3}{(n+1)(n+2)} = 3 \left( \frac{1}{n+1} - \frac{1}{n+2} \right)
   $$
   This telescoping nature means most terms will cancel out when summing.

2. **Finding the Sum as $$ n \to \infty $$:**
   $$
   \sum_{n=1}^{\infty} \frac{3}{(n+1)(n+2)} = 3 \left( \frac{1}{2} \right) = 1.5
   $$
   (Only the first term remains after cancellation.)

### Conclusion

- **Partial Sums:**
  - $$ S_1 = 0.5 $$
  - $$ S_2 = 0.75 $$
  - $$ S_3 = 0.9 $$
  - $$ S_4 = 1.0 $$
  - $$ S_5 \approx 1.0714 $$

- **Convergence:** The series is **convergent**.

- **Approximate Sum:** The series approaches an approximate sum of **1.5**.

**Answer to the First Partial Sum:**

$$ S_{1} = 0.5 $$
$$ S_{1} = 0.5 $$

Find the first five partial sums of the given series and determine whether the series appears to be
convergent or divergent. If it is convergent, find its approximate sum.

The first partial sum is .
(Type an integer or decimal rounded to four decimal places as needed.)

Upstudy AI Solution

Answer

Solution

Mind Expander

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Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. The first partial sum is . (Type an integer or decimal rounded to four decimal places as needed.)

Upstudy AI Solution

Answer

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Find the first five partial sums of the given series and determine whether the series appears to be
convergent or divergent. If it is convergent, find its approximate sum.

The first partial sum is .
(Type an integer or decimal rounded to four decimal places as needed.)