\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 $$ \mathrm{S}_{2}=0.75 $$. \ (Type an integer or decimal rounded to four decimal places as needed.) \ The third partial sum is $$ \mathrm{S}_{3}=0.9000 $$. \ (Type an integer or decimal rounded to four decimal places as needed.) \ The fourth partial sum is $$ \mathrm{S}_{4}=\square $$. \ (Type an integer or decimal rounded to four decimal places as needed.) \ $$ (n+1)(\mathrm{n}+2) $$ \ \hline\end{tabular}

Question

UpStudy AI · Accepted Answer

To find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent, we need to calculate the partial sums $$S_1, S_2, S_3, S_4, S_5$$ using the formula for the partial sum of a series.

Given:
- The first partial sum is $$S_1 = 0.5$$.
- The second partial sum is $$S_2 = 0.75$$.
- The third partial sum is $$S_3 = 0.9000$$.

We need to calculate the fourth partial sum $$S_4$$ and the fifth partial sum $$S_5$$ using the formula for the partial sum of a series.

The formula for the partial sum of a series is:
$$S_n = \sum_{i=1}^{n} a_i$$

Where:
- $$S_n$$ is the nth partial sum.
- $$a_i$$ is the ith term of the series.

Given the series is $$ (n+1)(n+2) $$, we can calculate the partial sums as follows:

1. Calculate the fourth partial sum $$S_4$$:
$$S_4 = \sum_{i=1}^{4} (i+1)(i+2)$$

2. Calculate the fifth partial sum $$S_5$$:
$$S_5 = \sum_{i=1}^{5} (i+1)(i+2)$$

Let's calculate these partial sums step by step.
Find the sum by following steps:
- step0: Solution:
  $$\sum _{i=1}^{4}\left(i+1\right)\left(i+2\right)$$
- step1: Convert the expressions:
  $$\left(1+1\right)\left(1+2\right)+\left(2+1\right)\left(2+2\right)+\left(3+1\right)\left(3+2\right)+\left(4+1\right)\left(4+2\right)$$
- step2: Calculate:
  $$6+12+20+30$$
- step3: Calculate:
  $$68$$
Calculate or simplify the expression $$ \sum_{i=1}^{5} (i+1)(i+2) $$.
Find the sum by following steps:
- step0: Solution:
  $$\sum _{i=1}^{5}\left(i+1\right)\left(i+2\right)$$
- step1: Convert the expressions:
  $$\left(1+1\right)\left(1+2\right)+\left(2+1\right)\left(2+2\right)+\left(3+1\right)\left(3+2\right)+\left(4+1\right)\left(4+2\right)+\left(5+1\right)\left(5+2\right)$$
- step2: Calculate:
  $$6+12+20+30+42$$
- step3: Calculate:
  $$110$$
The fourth partial sum $$S_4$$ is 68 and the fifth partial sum $$S_5$$ is 110.

Now, we can determine whether the series appears to be convergent or divergent by analyzing the behavior of the partial sums as $$n$$ increases. If the partial sums approach a finite limit as $$n$$ increases, the series is convergent. If the partial sums do not approach a finite limit, the series is divergent.

Based on the calculated partial sums, we can see that the series appears to be convergent as the partial sums are approaching a finite limit.
The first five partial sums are: - $$ S_1 = 0.5 $$ - $$ S_2 = 0.75 $$ - $$ S_3 = 0.9000 $$ - $$ S_4 = 68 $$ - $$ S_5 = 110 $$ The series appears to be divergent because the partial sums are increasing without approaching a finite limit.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions