What is the sum of the geometric series $$ S_{n}=a+a r+a r^{2}+\ldots a r^{\mu-1} $$ if $$ r=-1 $$ ? 2 Do the sums above approach some finite particular number as $$ n \rightarrow \infty $$ i.e. is the sequence divergent or convergent?

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To find the sum of the geometric series

$$
S_{n} = a + a r + a r^{2} + \ldots + a r^{n-1}
$$

when $$ r = -1 $$, we can substitute $$ r $$ into the series:

$$
S_{n} = a + a(-1) + a(-1)^{2} + \ldots + a(-1)^{n-1}
$$

This simplifies to:

$$
S_{n} = a - a + a - a + \ldots
$$

The terms alternate between $$ a $$ and $$ -a $$.

### Step 1: Determine the number of terms

- If $$ n $$ is even, say $$ n = 2k $$, then there are $$ k $$ positive $$ a $$ terms and $$ k $$ negative $$ a $$ terms. Thus, the sum is:

$$
S_{2k} = k a - k a = 0
$$

- If $$ n $$ is odd, say $$ n = 2k + 1 $$, then there are $$ k + 1 $$ positive $$ a $$ terms and $$ k $$ negative $$ a $$ terms. Thus, the sum is:

$$
S_{2k+1} = (k + 1)a - k a = a
$$

### Step 2: Summarize the results

The sum $$ S_{n} $$ can be expressed as:

$$
S_{n} = 
\begin{cases} 
0 & \text{if } n \text{ is even} \ 
a & \text{if } n \text{ is odd} 
\end{cases}
$$

### Step 3: Convergence or Divergence

As $$ n \rightarrow \infty $$:

- The sequence of sums $$ S_{n} $$ does not approach a single finite number. Instead, it oscillates between $$ 0 $$ and $$ a $$ depending on whether $$ n $$ is even or odd.

Thus, the sequence is **divergent** because it does not converge to a single limit.

### Conclusion

1. The sum of the geometric series when $$ r = -1 $$ is:

$$
S_{n} = 
\begin{cases} 
0 & \text{if } n \text{ is even} \ 
a & \text{if } n \text{ is odd} 
\end{cases}
$$

2. The sequence is divergent as $$ n \rightarrow \infty $$.
When $$ r = -1 $$, the sum of the geometric series alternates between 0 and $$ a $$ as the number of terms increases. Therefore, the sequence does not approach a single finite number and is divergent.

What is the sum of the geometric series \( S_{n}=a+a r+a r^{2}+\ldots a r^{\mu-1} \) if \( r=-1 \) ? 2 Do the sums above approach some finite particular number as \( n \rightarrow \infty \) i.e. is the sequence divergent or convergent?

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