Consider the following infinite series. Complete parts (a) through (c). a. Find the first four partial sums $$ \mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3} $$, and $$ \mathrm{S}_{4} $$. $$ \mathrm{S}_{1}=\square $$ (Simplify your answer. Type integers or decimals.) b. Find a formula for the nth partial sum $$ \mathrm{S}_{\mathrm{n}} $$ of the infinite series. $$ \mathrm{S}_{\mathrm{n}}=\square $$ (Use integers or fractions for any numbers in the expression.) Use the formula to find the next four partial sums $$ \mathrm{S}_{5}, \mathrm{~S}_{6}, \mathrm{~S}_{7} $$, and $$ \mathrm{S}_{8} $$ of the infinite series. $$ \mathrm{S}_{5}=\square $$ (Simplify your answer. Type integers or decimals.) c. Make a conjecture for the value of the series. $$ \mathrm{Sel}^{k} $$. answer box to complete your choice. A. The correct choice below and, if necessary, fill in the A. The value of the infinite series is (Type an integer or a simplified fraction.) B. The infinite series diverges.

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UpStudy AI · Accepted Answer

To solve the problem, we need to first identify the infinite series in question. However, since the series is not provided, I will assume a common type of series for demonstration purposes. Let's consider the infinite series:

$$
\sum_{n=1}^{\infty} \frac{1}{n^2}
$$

This is a well-known convergent series. Now, let's proceed with the steps outlined in the problem.

### Part (a): Find the first four partial sums $$ S_1, S_2, S_3, S_4 $$

The partial sums are calculated as follows:

$$
S_1 = \frac{1}{1^2} = 1
$$
$$
S_2 = S_1 + \frac{1}{2^2} = 1 + \frac{1}{4} = 1.25
$$
$$
S_3 = S_2 + \frac{1}{3^2} = 1.25 + \frac{1}{9} \approx 1.25 + 0.1111 \approx 1.3611
$$
$$
S_4 = S_3 + \frac{1}{4^2} = 1.3611 + \frac{1}{16} \approx 1.3611 + 0.0625 \approx 1.4236
$$

Thus, the first four partial sums are:
- $$ S_1 = 1 $$
- $$ S_2 = 1.25 $$
- $$ S_3 \approx 1.3611 $$
- $$ S_4 \approx 1.4236 $$

### Part (b): Find a formula for the nth partial sum $$ S_n $$

The formula for the nth partial sum of the series $$ \sum_{n=1}^{\infty} \frac{1}{n^2} $$ is given by:

$$
S_n = \sum_{k=1}^{n} \frac{1}{k^2}
$$

To find $$ S_5, S_6, S_7, S_8 $$, we can continue from $$ S_4 $$:

$$
S_5 = S_4 + \frac{1}{5^2} = 1.4236 + \frac{1}{25} = 1.4236 + 0.04 = 1.4636
$$
$$
S_6 = S_5 + \frac{1}{6^2} = 1.4636 + \frac{1}{36} \approx 1.4636 + 0.0278 \approx 1.4914
$$
$$
S_7 = S_6 + \frac{1}{7^2} = 1.4914 + \frac{1}{49} \approx 1.4914 + 0.0204 \approx 1.5118
$$
$$
S_8 = S_7 + \frac{1}{8^2} = 1.5118 + \frac{1}{64} \approx 1.5118 + 0.0156 \approx 1.5274
$$

Thus, the next four partial sums are:
- $$ S_5 \approx 1.4636 $$
- $$ S_6 \approx 1.4914 $$
- $$ S_7 \approx 1.5118 $$
- $$ S_8 \approx 1.5274 $$

### Part (c): Make a conjecture for the value of the series

The series $$ \sum_{n=1}^{\infty} \frac{1}{n^2} $$ converges to a known value:

$$
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.64493
$$

Thus, the conjecture for the value of the series is:

**A. The value of the infinite series is $$ \frac{\pi^2}{6} $$**.

If you have a specific series in mind, please provide it, and I can adjust the calculations accordingly.
**Part (a):** - $$ S_1 = 1 $$ - $$ S_2 = 1.25 $$ - $$ S_3 \approx 1.3611 $$ - $$ S_4 \approx 1.4236 $$ **Part (b):** $$ S_n = \sum_{k=1}^{n} \frac{1}{k^2} $$ - $$ S_5 \approx 1.4636 $$ - $$ S_6 \approx 1.4914 $$ - $$ S_7 \approx 1.5118 $$ - $$ S_8 \approx 1.5274 $$ **Part (c):** The infinite series converges to $$ \frac{\pi^2}{6} \approx 1.64493 $$.

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