Pregunta
Given the following telescoping series, determine whether it converges or diverges. If it converges, find its sum.
The series converges to 0 .
The series converges to 1 .
The series converges to
The series diverges.
The series converges to 0 .
The series converges to 1 .
The series converges to
The series diverges.
Ask by Pritchard Lowe. in the United States
Mar 24,2025
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The series converges to 0.
Solución
To determine whether the given telescoping series converges or diverges, we need to analyze the series term by term.
The given series is:
Let’s analyze the series term by term:
- For
:
- For
:
- For
:
Notice that the series is a telescoping series, meaning that each term cancels out the previous term except for the first and last terms.
The series can be rewritten as:
Since the series is a telescoping series, the sum of the series is the first term minus the limit of the last term as
The first term is
The limit of the last term as
Therefore, the sum of the series is
The series converges to 0.
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Simplifique esta solución Extra Insights
To tackle telescoping series, it’s often helpful to rewrite the terms in a way that shows cancellation more clearly. For your series, distributing terms can reveal that many intermediate terms cancel out, leading to a finite number of terms that do not cancel. This can be examined through the limit of its partial sums as n approaches infinity.
By carefully analyzing the behavior of the remaining terms as n becomes very large, you can confirm whether the series converges or diverges. If the leftover terms tend towards a specific value or remain bounded, it gives a strong hint toward convergence. On the contrary, if they keep growing or don’t settle, it indicates divergence.
If you’re keen on enhancing your understanding further, explore mathematical convergence tests such as the ratio test or comparison test. These tools often help distinguish between convergent and divergent series quickly! For those who enjoy a deeper dive, consider engaging with series within calculus textbooks or online platforms like Khan Academy, which provide exercises and visualizations to strengthen your grasp. Happy learning!
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