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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 S_{n} = 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 Σu_{n} (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 Σu_{n},

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 Σu_{n} 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 Σu_{n} 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 u_{n} = g(n)/h(n) where g(n) and h(n) are polynomials in n, then the
series Σu_{n} 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 Σu_{n} is convergent if

and

6. The root test. Let

Then the series Σu_{n} (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 Σu_{n} (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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