Math Problem Statement
What is the sum of the infinite geometric series? [ -3-\frac{3}{2}-\frac{3}{4}-\frac{3}{8}-\frac{3}{16}-\ldots ] ( -\frac{93}{16} ) ( -\frac{3}{32} ) -4 -6
Solution
The given series is:
This is an infinite geometric series, where the first term and the common ratio can be found by dividing the second term by the first term:
For a geometric series with , the sum to infinity is given by the formula:
Substituting the values and :
Thus, the sum of the infinite geometric series is:
Do you need further details or clarification on how this works?
Here are some related questions:
- How would the sum change if the common ratio were greater than 1?
- What happens to the sum of a geometric series if the common ratio is negative?
- How can you identify the terms of a geometric series from its formula?
- What is the sum of a geometric series with a common ratio of ?
- How do you calculate the partial sum of a geometric series?
Tip: When working with geometric series, always check that before applying the sum formula for infinity.
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Math Problem Analysis
Mathematical Concepts
Geometric Series
Infinite Series
Summation
Formulas
Sum of an infinite geometric series: \( S_\infty = \frac{a}{1 - r} \)
Theorems
Infinite Geometric Series Sum Theorem
Suitable Grade Level
Grades 9-12