Problem 15 Evaluate \(\sum_{k=1}^{\infty} \... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 7: Problem 15

Evaluate $\sum_{k=1}^{\infty} \frac{3}{7^{k}}$.

Short Answer

Expert verified

The sum of the infinite geometric series $\sum_{k=1}^{\infty} \frac{3}{7^{k}}$ is $\frac{1}{2}$.

Step by step solution

The first term (a) in the series is $\frac{3}{7}$ and the common ratio (r) is $\frac{1}{7}$. #Step 2: Verify that the series converges#

For an infinite geometric series to converge (have a finite sum), the common ratio must be between -1 and 1. In this case, the common ratio is $\frac{1}{7}$, which falls between -1 and 1, so the series converges. #Step 3: Apply the geometric series sum formula#

Now, we'll apply the geometric series sum formula: $S = \frac{a}{1 - r}$. We already know the values of a and r, so we just need to substitute them into the formula: $S = \frac{\frac{3}{7}}{1 - \frac{1}{7}}$ #Step 4: Simplify the expression#

To simplify the expression, we'll first simplify the denominator: $1 - \frac{1}{7} = \frac{7 - 1}{7} = \frac{6}{7}$ Now, we can substitute the simplified denominator back into the expression: $S = \frac{\frac{3}{7}}{\frac{6}{7}}$ To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction: $S = \frac{3}{7} \times \frac{7}{6}$ The 7 in the numerator and denominator cancel each other out: $S = \frac{3}{6}$ Lastly, we can simplify S to its lowest terms: $S = \frac{1}{2}$ So, the sum of the given infinite geometric series is $\frac{1}{2}$.

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91影视

Evaluate \(\sum_{k=1}^{\infty} \frac{3}{7^{k}}\).

Short Answer

Step by step solution

The first term (a) in the series is \(\frac{3}{7}\) and the common ratio (r) is \(\frac{1}{7}\). #Step 2: Verify that the series converges#

Now, we'll apply the geometric series sum formula: \(S = \frac{a}{1 - r}\). We already know the values of a and r, so we just need to substitute them into the formula: \(S = \frac{\frac{3}{7}}{1 - \frac{1}{7}}\) #Step 4: Simplify the expression#

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