Compute each sum below. Give exact values, not decimal approximations. If the sum does not exist, click on "No sum". $$ 4\left(-\frac{1}{4}\right)+4\left(-\frac{1}{4}\right)^{2}+4\left(-\frac{1}{4}\right)^{3}+\ldots=\prod $$ $$ \sum_{k=1}^{\infty} 2\left(-\frac{3}{5}\right)^{k}=\square $$

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To compute the sums, we will analyze each series separately.

1. **First Sum:**
   $$
   4\left(-\frac{1}{4}\right) + 4\left(-\frac{1}{4}\right)^{2} + 4\left(-\frac{1}{4}\right)^{3} + \ldots
   $$
   This series can be recognized as a geometric series where the first term $$ a = 4\left(-\frac{1}{4}\right) = -1 $$ and the common ratio $$ r = -\frac{1}{4} $$.

The formula for the sum of an infinite geometric series is given by:
   $$
   S = \frac{a}{1 - r}
   $$
   provided that $$ |r| < 1 $$.

Here, $$ |r| = \left| -\frac{1}{4} \right| = \frac{1}{4} < 1 $$, so we can apply the formula:
   $$
   S = \frac{-1}{1 - \left(-\frac{1}{4}\right)} = \frac{-1}{1 + \frac{1}{4}} = \frac{-1}{\frac{5}{4}} = -\frac{4}{5}
   $$

Thus, the first sum is:
   $$
   -\frac{4}{5}
   $$

2. **Second Sum:**
   $$
   \sum_{k=1}^{\infty} 2\left(-\frac{3}{5}\right)^{k}
   $$
   This is also a geometric series where the first term $$ a = 2\left(-\frac{3}{5}\right) = -\frac{6}{5} $$ and the common ratio $$ r = -\frac{3}{5} $$.

Again, we check the condition for convergence:
   $$
   |r| = \left| -\frac{3}{5} \right| = \frac{3}{5} < 1
   $$
   Therefore, we can use the sum formula:
   $$
   S = \frac{a}{1 - r} = \frac{-\frac{6}{5}}{1 - \left(-\frac{3}{5}\right)} = \frac{-\frac{6}{5}}{1 + \frac{3}{5}} = \frac{-\frac{6}{5}}{\frac{8}{5}} = -\frac{6}{8} = -\frac{3}{4}
   $$

Thus, the second sum is:
   $$
   -\frac{3}{4}
   $$

In conclusion, the results for the sums are:
1. $$ -\frac{4}{5} $$
2. $$ -\frac{3}{4} $$
1. $$ -\frac{4}{5} $$ 2. $$ -\frac{3}{4} $$

Compute each sum below.
Give exact values, not decimal approximations.
If the sum does not exist, click on “No sum”.

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Compute each sum below. Give exact values, not decimal approximations. If the sum does not exist, click on “No sum”.