%5En%2Fn-converge-or-diverge-sequence.jpeg "(-1)^n/n Converge Or Diverge Sequence")
The Convergence of (-1)^n/n: A Look at the Alternating Series Test
The sequence (-1)^n/n is a classic example of an alternating series, meaning its terms alternate in sign. Understanding whether this series converges or diverges requires a specific test: the Alternating Series Test.
The Alternating Series Test
The Alternating Series Test states that an alternating series converges if:
- The absolute value of the terms decreases monotonically: This means the terms are getting smaller in absolute value as 'n' increases.
- The limit of the terms approaches zero: This means the terms eventually get arbitrarily close to zero.
Let's apply this test to our sequence (-1)^n/n:
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Decreasing Terms: The absolute value of the terms is 1/n. As 'n' increases, 1/n decreases, satisfying the first condition.
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Limit of Terms: The limit as 'n' approaches infinity of 1/n is zero. This satisfies the second condition.
Conclusion
Since the sequence (-1)^n/n satisfies both conditions of the Alternating Series Test, we can conclude that it converges.
Importance of Convergence
The convergence of an alternating series implies that the sum of its infinite terms approaches a finite value. This has applications in various areas, including:
- Calculus: Understanding convergence is crucial in approximating functions and solving differential equations.
- Statistics: Convergence is used in statistical models for estimating probabilities and analyzing data.
- Physics: Convergent series are used to model physical phenomena, such as wave propagation and heat transfer.
Visualizing Convergence
To visualize the convergence of the sequence, consider plotting the first few terms:
(-1)^1/1 = -1 (-1)^2/2 = 1/2 (-1)^3/3 = -1/3 (-1)^4/4 = 1/4 ...
You'll see the terms oscillating between positive and negative values, getting closer and closer to zero. This visual representation helps understand the concept of convergence.
In conclusion, the sequence (-1)^n/n converges due to the Alternating Series Test. This convergence plays a significant role in various mathematical and scientific disciplines.