\begin{tabular}{l} 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 $$ \mathrm{S}_{1}=0.5 $$. \ (Type an integer or decimal rounded to four decimal places as needed.) \ The second partial sum is $$ \mathrm{S}_{2}=\square $$ \ (Type an integer or decimal rounded to four decimal places as needed.) \ \hline$$ (\mathrm{n}+1)(\mathrm{n}+2) $$ \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

Certainly! Let's analyze the given series and find its first five partial sums to determine its convergence.

### **Given Series:**
$$
\sum_{n=1}^{\infty} \frac{1}{n(n+1)}
$$

### **Partial Sums Calculation:**

The partial sum $$ S_k $$ for the first $$ k $$ terms of the series is given by:
$$
S_k = \sum_{n=1}^{k} \frac{1}{n(n+1)}
$$

This can be simplified using partial fraction decomposition:
$$
\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}
$$

Thus, the partial sum becomes:
$$
S_k = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{k} - \frac{1}{k+1}\right)
$$

This telescopes to:
$$
S_k = 1 - \frac{1}{k+1}
$$

Now, let's compute the first five partial sums:

1. **First Partial Sum ($$ S_1 $$):**
$$
S_1 = 1 - \frac{1}{1+1} = 1 - \frac{1}{2} = 0.5
$$

2. **Second Partial Sum ($$ S_2 $$):**
$$
S_2 = 1 - \frac{1}{2+1} = 1 - \frac{1}{3} \approx 0.6667
$$

3. **Third Partial Sum ($$ S_3 $$):**
$$
S_3 = 1 - \frac{1}{3+1} = 1 - \frac{1}{4} = 0.75
$$

4. **Fourth Partial Sum ($$ S_4 $$):**
$$
S_4 = 1 - \frac{1}{4+1} = 1 - \frac{1}{5} = 0.8
$$

5. **Fifth Partial Sum ($$ S_5 $$):**
$$
S_5 = 1 - \frac{1}{5+1} = 1 - \frac{1}{6} \approx 0.8333
$$

### **Summary of Partial Sums:**

$$
\begin{array}{|c|c|}
\hline
\text{Partial Sum} & \text{Value} \
\hline
S_1 & 0.5000 \
S_2 & 0.6667 \
S_3 & 0.7500 \
S_4 & 0.8000 \
S_5 & 0.8333 \
\hline
\end{array}
$$

### **Convergence Analysis:**

From the partial sums, we observe that as $$ k $$ increases, $$ S_k $$ approaches 1. Formally:
$$
\lim_{k \to \infty} S_k = \lim_{k \to \infty} \left(1 - \frac{1}{k+1}\right) = 1
$$

**Conclusion:** The series **converges** to **1**.
The first five partial sums are: - $$ S_1 = 0.5000 $$ - $$ S_2 \approx 0.6667 $$ - $$ S_3 = 0.7500 $$ - $$ S_4 = 0.8000 $$ - $$ S_5 \approx 0.8333 $$ The series converges to **1**.

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