Math Problem Statement
Solution
Let's walk through the problem to determine if the given series converges or diverges, and if it converges, find the sum.
Given:
Step 1: Rewrite the series
The term inside the summation can be split:
This simplifies to:
Step 2: Analyze the two series
Both components are geometric series:
-
For :
- This is a geometric series with . Since , it converges.
- Sum of the series:
-
For :
- This is another geometric series with . Since , it also converges.
- Sum of the series:
Step 3: Find the total sum
Since both geometric series converge, the total sum is the sum of their individual sums:
Step 4: Conclusion
The given series converges and its sum is .
Correct Answer:
The correct choice is A:
- "The series converges because it is the sum of two geometric series, each with ."
- The sum of the series is:
Do you have any questions, or would you like further clarification?
Here are five related questions for further exploration:
- What are the conditions for a geometric series to converge?
- How would the sum change if one of the terms had ?
- Can you generalize the sum of a series like ?
- What is the relationship between convergence tests like the Ratio Test and geometric series?
- How does changing the base of the exponents (e.g., or ) affect convergence?
Tip: Always check if you can split complex series into simpler parts, such as sums of geometric series, to analyze convergence easily.
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Math Problem Analysis
Mathematical Concepts
Series
Geometric Series
Convergence
Summation
Formulas
Geometric series formula: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}\) for |r| < 1
Theorems
Geometric Series Convergence Theorem
Suitable Grade Level
College-level mathematics or advanced high school
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