Show all your work to receive full credit. Write your answers as complete sentences.

Given the following expression:

a) Using sigma notation, write the expression as an infinite series.
b) Find the partial sum .
c) Determine if the series is convergent or divergent. If it’s convergent, find its sum.

Ask by Reyes Coleman. in the United States

Mar 24,2025

Question

Show all your work to receive full credit. Write your answers as complete sentences.

Given the following expression:

a) Using sigma notation, write the expression as an infinite series.
b) Find the partial sum .
c) Determine if the series is convergent or divergent. If it’s convergent, find its sum.

Ask by Reyes Coleman. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

1. We start by recognizing that the given expression  
   $$
   \frac{3}{10}+\frac{9}{100}+\frac{27}{1000}+\frac{81}{10000}+\cdots
   $$  
   is a geometric series whose first term is  
   $$
   a=\frac{3}{10}
   $$  
   and whose common ratio is  
   $$
   r=\frac{3}{10}
   $$  
   since each term can be obtained by multiplying the previous term by $$\frac{3}{10}$$.

2. **(a) Writing the series in sigma notation:**  
   Because the first term is $$\frac{3}{10}$$ and each subsequent term is found by multiplying by $$\frac{3}{10}$$, we can write the series as  
   $$
   \sum_{n=1}^{\infty}\left(\frac{3}{10}\right)^n.
   $$  
   Alternatively, by starting the index at 0, it can also be written as  
   $$
   \sum_{n=0}^{\infty}\left(\frac{3}{10}\right)^{n+1}.
   $$

3. **(b) Finding the partial sum $$S_{5}$$:**  
   The formula for the sum of the first $$n$$ terms of a geometric series is  
   $$
   S_n=\frac{a\left(1-r^n\right)}{1-r}.
   $$  
   For $$n=5$$, we have  
   $$
   S_5=\frac{\frac{3}{10}\left(1-\left(\frac{3}{10}\right)^5\right)}{1-\frac{3}{10}}.
   $$
   First, compute $$\left(\frac{3}{10}\right)^5$$:  
   $$
   \left(\frac{3}{10}\right)^5=\frac{3^5}{10^5}=\frac{243}{100000}.
   $$
   The denominator simplifies as follows:  
   $$
   1-\frac{3}{10}=\frac{7}{10}.
   $$
   Now, substitute these values into the formula:  
   $$
   S_5=\frac{\frac{3}{10}\left(1-\frac{243}{100000}\right)}{\frac{7}{10}}.
   $$
   Dividing by $$\frac{7}{10}$$ is equivalent to multiplying by $$\frac{10}{7}$$, so we have  
   $$
   S_5=\frac{3}{10}\left(1-\frac{243}{100000}\right)\cdot\frac{10}{7}=\frac{3}{7}\left(1-\frac{243}{100000}\right).
   $$
   Expressing the term inside the parentheses as a single fraction:  
   $$
   1-\frac{243}{100000}=\frac{100000-243}{100000}=\frac{99757}{100000}.
   $$
   Thus, the partial sum is  
   $$
   S_5=\frac{3}{7}\cdot\frac{99757}{100000}=\frac{3\times99757}{7\times100000}.
   $$

4. **(c) Determining convergence and finding the sum of the infinite series:**  
   An infinite geometric series converges if and only if $$|r|<1$$. Here,  
   $$
   |r|=\left|\frac{3}{10}\right|=0.3<1,
   $$  
   so the series converges. The sum of the infinite series is given by the formula  
   $$
   S_{\infty}=\frac{a}{1-r}.
   $$
   Substituting $$a=\frac{3}{10}$$ and $$r=\frac{3}{10}$$ gives  
   $$
   S_{\infty}=\frac{\frac{3}{10}}{1-\frac{3}{10}}=\frac{\frac{3}{10}}{\frac{7}{10}}=\frac{3}{7}.
   $$

In summary, the answers are as follows:

1. (a) The series in sigma notation is  
   $$
   \sum_{n=1}^{\infty}\left(\frac{3}{10}\right)^n.
   $$

2. (b) The partial sum $$S_{5}$$ is  
   $$
   S_5=\frac{3}{7}\left(1-\frac{243}{100000}\right)=\frac{3\times99757}{700000}.
   $$

3. (c) The series is convergent, and its sum is  
   $$
   \frac{3}{7}.
   $$
The series can be written as $$ \sum_{n=1}^{\infty}\left(\frac{3}{10}\right)^n $$. The partial sum $$ S_5 $$ is $$ \frac{3}{7}\left(1-\frac{243}{100000}\right) $$. The series is convergent and its sum is $$ \frac{3}{7} $$.

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