Website owner: James Miller
CONVERGENCE TESTS, COMPARISON TEST, RATIO TEST, INTEGRAL TEST, POLYNOMIAL TEST, RAABE’S TEST
Given a particular series the first question one wishes to answer is whether the series converges or not. There is no single universal test that one can use to determine whether a series converges. Instead, there are a number of tests, some of which may be of use in one case, others in another.
Fundamental Principle. If a sequence Sn = φ(n) always increases as n increases but always remains less than a fixed number Q, then exists and is not greater than Q.
Convergence tests
1. Comparison Test.
Convergence. A positive series is convergent if each of its terms is less than or equal to the corresponding terms of a series that is known to be convergent.
Divergence. A positive series is divergent if each of its terms is greater than or equal to the corresponding terms of a series that is known to be divergent.
2. Ratio Test. Let
Then the series Σun (positive or mixed-term)
● converges (absolutely) if L < 1
● diverges if L > 1
If L = 1 the test fails.
If the test fails try the test of the following theorem.
Theorem 1. If, for a given series Σun,
the series converges if b - a > 1 and diverges if b - a 1.
Example. Test the series
for convergence.
Solution. Using the ratio test
Thus the test is inconclusive.
Using Theorem 1,
and b - a = 3/2 - 1/2 = 1
Therefore, by the theorem, the series diverges.
3. Integral Test. Let the general term of the series Σun be f(n), and let f(x) be the function obtained by replacing n by the continuous variable x. Now if for all values of x > a (where a is a positive integer) this function f(x) is positive and decreasing and if f(x) → 0 as x → ∞, then the series Σun converges if the integral
is convergent and diverges if this integral is divergent.
Example. Consider the series
Here
and
For all positive values of x this function is positive and decreasing, and as x → ∞, f(x) → 0 . In addition
Thus the integral converges and the series is convergent.
4. Polynomial test. If un = g(n)/h(n) where g(n) and h(n) are polynomials in n, then the series Σun is convergent if the degree of h(n) exceeds that of g(n) by more than 1; otherwise the series is divergent.
Example. Consider the series
Here
The degree of the denominator exceeds that of the numerator by 2 and therefore the series is convergent.
5. Alternating series test. An alternating series Σun is convergent if
and
6. The root test. Let
Then the series Σun (positive or mixed-term)
● converges (absolutely) if L < 1
● diverges if L > 1
If L = 1 the test fails.
7. Raabe’s test. Let
Then the series Σun (positive or mixed-term)
● converges (absolutely) if L > 1
● diverges or converges conditionally if L < 1
If L = 1 the test fails.
Raabe’s test is often used when the ratio test fails.
References
Middlemiss. Differential and Integral Calculus.
Ayres. Calculus (Schaum).
Oakley. The Calculus (COS).
Spiegel. Advanced Calculus (Schaum).
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