Math Problem Statement
Solution
The series presented is:
We can start by simplifying the term inside the summation.
This expression can be decomposed using partial fraction decomposition as follows:
To find and , we solve the equation:
Multiplying both sides by gives:
Expanding the right-hand side:
Now, by comparing coefficients of and the constant terms, we get the following system of equations:
From equation (2), we have . Substituting into equation (1), we get:
So the partial fraction decomposition is:
Now, we substitute this into the original summation:
This is a telescoping series, where most terms cancel out. Let's expand the first few terms to see the cancellation:
All intermediate terms cancel out, and we're left with:
Thus, the sum of the series is:
Do you have any questions or would you like more details on any step?
Here are 5 related questions to expand on this:
- What is a telescoping series, and how does it simplify the summation process?
- Can you provide another example where partial fraction decomposition is used in summing a series?
- How would you approach the series if the lower bound of the summation was different?
- Why do most terms cancel in a telescoping series, and can you give an example where they do not?
- How can this method of solving be applied to summing infinite series with higher powers in the denominator?
Tip: When dealing with rational expressions in series, always check if partial fraction decomposition can simplify the problem—it often turns a complex sum into a telescoping series!
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Math Problem Analysis
Mathematical Concepts
Series
Partial Fraction Decomposition
Telescoping Series
Formulas
Partial Fraction Decomposition: 5 / (k^2 - k) = 5 / [k(k-1)] = A / (k-1) + B / k
Sum of Telescoping Series: Terms cancel out to leave a few remaining values
Theorems
Telescoping Series Theorem: In a series where successive terms cancel out, the sum is determined by the non-cancelled terms.
Suitable Grade Level
Grades 11-12
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