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

Ask by Morgan Lyons. 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.)
The second partial sum is .
(Type an integer or decimal rounded to four decimal places as needed.)

Ask by Morgan Lyons. in the United States

Jan 12,2025

UpStudy AI · Accepted Answer

To find the first five partial sums of the series and determine its convergence, let's analyze the series:

$$
\sum_{n=1}^{\infty} \frac{3}{(n+1)(n+2)}
$$

**Step 1: Partial Fraction Decomposition**

First, we can express the general term using partial fractions:

$$
\frac{3}{(n+1)(n+2)} = 3\left(\frac{1}{n+1} - \frac{1}{n+2}\right)
$$

This decomposition shows that the series is telescoping.

**Step 2: Sum of Partial Sums**

The $$ N $$-th partial sum $$ S_N $$ can be written as:

$$
S_N = 3\left(\frac{1}{2} - \frac{1}{N+2}\right)
$$

This simplifies to:

$$
S_N = \frac{3}{2} - \frac{3}{N+2}
$$

**Step 3: Calculating the First Five Partial Sums**

Given that the first partial sum $$ S_1 = 0.5 $$, let's compute the next partial sums using the formula:

1. **Second Partial Sum ($$ S_2 $$)**:
   $$
   S_2 = \frac{3}{2} - \frac{3}{4} = 0.75
   $$

2. **Third Partial Sum ($$ S_3 $$)**:
   $$
   S_3 = \frac{3}{2} - \frac{3}{5} = 0.9
   $$

3. **Fourth Partial Sum ($$ S_4 $$)**:
   $$
   S_4 = \frac{3}{2} - \frac{3}{6} = 1.0
   $$

4. **Fifth Partial Sum ($$ S_5 $$)**:
   $$
   S_5 = \frac{3}{2} - \frac{3}{7} \approx 1.0714
   $$

**Step 4: Determining Convergence**

As $$ N $$ approaches infinity, the term $$ \frac{3}{N+2} $$ approaches 0. Therefore, the series converges to:

$$
\lim_{N \to \infty} S_N = \frac{3}{2} = 1.5
$$

**Conclusion**

The series is **convergent**, and its approximate sum is **1.5**.

**Answer for the Second Partial Sum:**

$$
\mathrm{S}_{2} = 0.75
$$
The second partial sum is $$ S_2 = 0.75 $$.