Math Problem Statement
Consider the finite geometric series. β π
1 π 4 ( 1 3 ) π β 1 k=1 β n β 4( 3 1 β ) kβ1
Evaluate the series for each of the given values of π nβ. You may round your answers to five decimal places.
π
6 n=6.β Preview β π
8 n=8β. Preview β π
10 n=10β.
Solution
To evaluate the given finite geometric series for different values of , we can use the formula for the sum of a geometric series:
where:
- is the first term of the series,
- is the common ratio, and
- is the number of terms.
For the series , the first term is 4 (when ) and the common ratio is .
Let's calculate the sum for each value of .
-
For :
-
For :
-
For :
Let's calculate these values.Here are the evaluated sums for the geometric series with different values of , rounded to five decimal places:
- For :
- For :
- For :
Would you like any further details or explanations?
Here are some related questions to consider:
- How does the common ratio affect the convergence of a geometric series?
- What happens to the sum as approaches infinity for this series?
- How would the sum change if the first term were different?
- What is the significance of the common ratio being less than 1?
- Can a geometric series have a negative common ratio? If so, how does it affect the series?
- What if the common ratio was greater than 1? How would that affect the series?
- How can we use geometric series in real-world applications?
- What are the conditions for a geometric series to converge?
Tip: Understanding the behavior of a geometric series as the number of terms increases can help in approximating the sum for large .
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Math Problem Analysis
Mathematical Concepts
Geometric Series
Formulas
Sum of a geometric series: \( S_n = a \frac{1 - r^n}{1 - r} \)
Theorems
-
Suitable Grade Level
Advanced High School
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