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Poirot1
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Find the sum to infinity of the series $1 +2z +3z^2+4z^3+...$
What is the sum to infinity of this nearly geometric series?
- Context: MHB
- Thread starter Poirot1
- Start date
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SUMMARY
The sum to infinity of the series \(1 + 2z + 3z^2 + 4z^3 + \ldots\) can be derived using calculus techniques. By integrating the series and then differentiating the resulting function, the closed form of the sum is established as \(\frac{1}{(1-z)^2}\) for \(|z| < 1\). This approach leverages the properties of power series and their convergence criteria.
PREREQUISITES- Understanding of power series and their convergence
- Basic calculus, including differentiation and integration
- Familiarity with geometric series and their sums
- Knowledge of complex numbers and their properties
- Study the derivation of the geometric series sum formula
- Learn about the properties of power series and their convergence
- Explore advanced calculus techniques for series manipulation
- Investigate applications of series in complex analysis
Mathematicians, students studying calculus, and anyone interested in series convergence and manipulation techniques.
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