QQuestionMathematics
QuestionMathematics
Determine the sum of the following series.
\sum_{n= 1}^{\infty} \frac{(- 1)^{n- 1}}{5^{n}}
12 months agoReport content
Answer
Full Solution Locked
Sign in to view the complete step-by-step solution and unlock all study resources.
Step 1:: Recognize that this is an alternating series with terms in the form of \frac{(- 1)^{n- 1}}{5^{n}}.
Step 2:: To find the sum of an alternating series, we can use the alternating series test.
The test states that if the limit as n approaches infinity of |a\_n| is zero, then the sum of the series is convergent and is equal to the limit as n approaches infinity of the sum from 1 to n of (- 1)^(n+ 1) * a\_n.
Step 3:: In this case, |a\_n| is simply \frac{1}{5^{n}}.
We know that the limit as n approaches infinity of $$\frac{1}{5^{n}}$$ is 0, since 5 raised to any power greater than 0 will approach infinity and 1 divided by infinity is 0.
Step 4:: Now we can apply the formula from the alternating series test:
\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{5^{n}}
Step 5:: We can simplify this expression by combining the constants (- 1)^(n+ 1) and \frac{1}{5^{n}} into a single fraction:
\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{5^{n}} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{5^{n}}
Step 6:: To find the sum of this series, we can use the formula for the sum of an infinite geometric series, which is \frac{a}{1 -r}, where a is the first term and r is the common ratio.
In this case, a = $$\frac{1}{5}$$ and r = -$$\frac{1}{5}$$.
Step 7:: Plugging these values into the formula, we get:
\frac{\frac{1}{5}}{1 - (-\frac{1}{5})} = \frac{\frac{1}{5}}{\frac{6}{5}} = \frac{1}{6} \approx 0.1667
Final Answer
The sum of the series \sum_{n= 1}^{\infty} \frac{(- 1)^{n- 1}}{5^{n}} is approximately 0.1667.
Need Help with Homework?
Stuck on a difficult problem? We've got you covered:
- Post your question or upload an image
- Get instant step-by-step solutions
- Learn from our AI and community of students