The Comparison Test is a method used to determine the convergence or divergence of infinite series or sequences by comparing them to another series or sequence whose convergence behavior is already known. This test simplifies the process of analyzing series by establishing a relationship between two sequences or series, allowing us to conclude the behavior of one based on the other.
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For the Comparison Test, if you have two positive series $$ ext{a}_n$$ and $$ ext{b}_n$$ and $$ ext{a}_n \leq ext{b}_n$$ for all n large enough, then if $$ ext{b}_n$$ converges, so does $$ ext{a}_n$$.
Conversely, if $$ ext{a}_n \geq ext{b}_n$$ and $$ ext{b}_n$$ diverges, then $$ ext{a}_n$$ also diverges.
The Comparison Test can be applied only when both series involved have non-negative terms, making it essential to check the terms before using the test.
This test is particularly useful for series involving polynomial, exponential, and logarithmic functions where direct evaluation of convergence can be complex.
The Limit Comparison Test provides an alternative approach where instead of direct comparison, the limit of the ratio of terms from two series is examined to draw conclusions about their convergence.
Review Questions
How does the Comparison Test help in determining the convergence of a series, and what conditions must be met for its application?
The Comparison Test helps by allowing us to assess whether one series behaves similarly to another whose convergence is known. The key conditions include that both series must have positive terms and that one must be less than or equal to the other for sufficiently large n. If the larger series converges, then so does the smaller one; similarly, if the larger diverges, so does the smaller.
Illustrate with an example how the Comparison Test can be applied to establish the convergence of a particular infinite series.
Consider the series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ which is known to converge (it’s a p-series with p=2). To use the Comparison Test on the series $$\sum_{n=1}^{\infty} \frac{1}{n^2 + n}$$, we can see that for all n, $$\frac{1}{n^2 + n} \leq \frac{1}{n^2}$$. Since $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ converges, it follows that $$\sum_{n=1}^{\infty} \frac{1}{n^2 + n}$$ also converges by the Comparison Test.
Evaluate and discuss potential limitations when using the Comparison Test versus other convergence tests available for infinite series.
While the Comparison Test is powerful, it has limitations, such as only being applicable to positive term series. In cases where terms are negative or oscillating, other tests like the Ratio or Root Tests may be more appropriate. Additionally, sometimes finding an appropriate series for comparison can be difficult. This highlights the importance of understanding various convergence tests and knowing when each is best suited for a given problem.
The property of a sequence or series that does not approach a finite limit, often characterized by growing indefinitely or oscillating without settling.