There are different ways of testing if a series converges or diverges. One of the methods used in calculus is the Integral Test. This test involves finding an integral expression for the terms of the series and then evaluating it to determine if the series converges or diverges.
Integral Test for Series
The integral test can only be applied to a series whose terms are positive and decreasing. If the series satisfies these conditions, we can find an integral expression for the terms of the series as follows:
an = f(n)
where f(x) is a continuous, positive, and decreasing function for x >= k, where k is a positive integer.
Then, we can evaluate the integral of f(x) from k to infinity, as follows:
integral[f(x)]infinityk = L
If L is finite and positive, then the series converges. If L is infinite or undefined, then the series diverges.
Example
For example, consider the series:
1 + 1/4 + 1/9 + 1/16 + ...
Here, the terms are decreasing and positive. We can find an integral expression for the terms as follows:
an = 1/n2
We can evaluate the integral of f(x) from 1 to infinity:
integral[1/x2]infinity1 = [(-1/ x)]infinity1 = 1
Since the integral is finite and positive, we can conclude that the series converges.
The integral test is a powerful tool for testing convergence or divergence of infinite series. However, it should be used with caution as it may not always work for all series.
It is important to note that there are other methods for testing convergence or divergence of series, such as the Comparison Test, Ratio Test and Root Test. Therefore, it is recommended to apply different tests to check if the results are consistent.
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