Strategy for Testing Series

We now have several ways of testing a series for convergence or divergence; the problem

is to decide which test to use on which series. In this respect testing series is similar to integrating

functions. Again there are no hard and fast rules about which test to apply to a

given series, but you may find the following advice of some use.

It is not wise to apply a list of the tests in a specific order until one finally works. That

would be a waste of time and effort. Instead, as with integration, the main strategy is to

classify the series according to its form.

1. If the series is of the form , it is a -series, which we know to be convergent

if and divergent if .

2. If the series has the form or , it is a geometric series, which converges

if and diverges if . Some preliminary algebraic manipulation

may be required to bring the series into this form.

3. If the series has a form that is similar to a -series or a geometric series, then

one of the comparison tests should be considered. In particular, if is a rational

function or algebraic function of (involving roots of polynomials), then the

series should be compared with a -series. (The value of should be chosen as

in Section 8.3 by keeping only the highest powers of in the numerator and

denominator.) The comparison tests apply only to series with positive terms, but

if has some negative terms, then we can apply the Comparison Test to

and test for absolute convergence.

4. If you can see at a glance that , then the Test for Divergence

should be used.

5. If the series is of the form or , then the Alternating Series

Test is an obvious possibility.

6. Series that involve factorials or other products (including a constant raised to the

th power) are often conveniently tested using the Ratio Test. Bear in mind that

as for all -series and therefore all rational or algebraic

functions of . Thus, the Ratio Test should not be used for such series.

7. If , where is easily evaluated, then the Integral Test is effective

(assuming the hypotheses of this test are satisfied).

Convergence Tests

Absolute Convergence

If the series  |a_n| converges, then the series  a_n also converges.

Alternating Series Test

If for all n, a_n is positive, non-increasing (i.e. 0 < a_n+1 <= a_n), and approaching zero, then the alternating series
 (-1)ⁿ a_n   and    (-1)^n-1 a_n
both converge.
If the alternating series converges, then the remainder R_N = S - S_N (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is bounded by |R_N| <= a_N+1

Deleting the first N Terms

If N is a positive integer, then the series

an and

an   n=N+1

both converge or both diverge.

Direct Comparison Test

If 0 <= a_n <= b_n for all n greater than some positive integer N, then the following rules apply:
If  b_n converges, then  a_n converges.
If  a_n diverges, then  b_n diverges.

Geometric Series Convergence

The geometric series is given by
 a rⁿ = a + a r + a r² + a r³ + ...
If |r| < 1 then the following geometric series converges to a / (1 - r).
If |r| >= 1 then the above geometric series diverges.

Integral Test

If for all n >= 1, f(n) = a_n, and f is positive, continuous, and decreasing then  

an and

an

either both converge or both diverge.
If the above series converges, then the remainder R_N = S - S_N (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is bounded by 0< = R_N <= (N..) f(x) dx.

Limit Comparison Test

If lim (n-->) (a_n / b_n) = L,
where a_n, b_n > 0 and L is finite and positive,
then the series  a_n and  b_n either both converge or both diverge.

n^th-Term Test for Divergence

If the sequence {a_n} does not converge to zero, then the series  a_ndiverges.

p-Series Convergence

The p-series is given by
 1/n^p = 1/1^p + 1/2^p + 1/3^p + ...
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.

Ratio Test

If for all n, n  0, then the following rules apply:
Let L = lim (n -- > ) | a_n+1 / a_n |.
If L < 1, then the series  a_n converges.
If L > 1, then the series  a_n diverges.
If L = 1, then the test in inconclusive.

Root Test

Let L = lim (n -- > ) | a_n |^1/n.
If L < 1, then the series  a_n converges.
If L > 1, then the series  a_n diverges.
If L = 1, then the test in inconclusive.

Taylor Series Convergence

If f has derivatives of all orders in an interval I centered at c, then the Taylor series converges as indicated:
 (1/n!) f⁽ⁿ⁾(c) (x - c)ⁿ = f(x)
if and only if lim (n-->) RN = 0 for all x in I.
The remainder R_N = S - S_N of the Taylor series (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is equal to (1/(n+1)!) f⁽ⁿ⁺¹⁾(z) (x - c)ⁿ⁺¹, where z is some constant between x and c.

Reference: Math.Com@ http://www.math.com/tables/expansion/tests.htm

L-R Paulzy......

November 10th.

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