Integral

If s_n = , then look at the integral that “matches” it. Whether converges or diverges is also the result for the series. In this case, since the integral diverges, so does the series. This can be understood by realizing that graphically the integral is below the series, and the comparison test for improper integrals says that the series would diverge, also. In the drawing, the rectangles for the Riemann sum represent the series, and the area below the curve represents the integral.

This can be done with any series and its associated integral.

p-series

s_n = converges if p > 1 (just like converges when p > 1)

ratio

A series converges if the ratio lim < 1.

A series diverges if the ratio lim > 1.

A series either converges or diverges if the ratio lim = 1.

(inconclusive)

ex: If s_n = , then a_n =

The ratio test says that = =  < 1 , so s_n converges.

Note: s_n is geometric with a₁ = and r = .

So s_n = = = , so s_n converges to .

root

A series s_n converges if < 1 ( and diverges if the root is greater than 1, and is inconclusive if the root is equal to one, just like the ratio test)

Ex: s_n = , so a_n =

The root test says that =  < 1 so s_n converges.

Note: lim = 1 (Use logarithms to prove this!)

Ex: s_n = so a_n =

Then the root test says = = -> 0 so the series converges.

3. Alternating series

Alternating Series

An alternating series is a series whose terms are alternately positive and negative.

Examples:

In general, just knowing that tells us very little about the convergence of the

series ; however, it turns out that an alternating series must converge if the terms have a limit of 0 and the terms decrease in magnitude.

Alternating Series Test

If , then an alternating series

converges if both of the following conditions are satisfied:

1)

2) is a decreasing sequence; that is, for all n.

Note: This does not say that if , the series diverges by the Alternating Series

Test. The Alternating Series Test can only be used to prove convergence.

If , then the series diverges by the nth Term Test for Divergence,

not by the Alternating Series Test.

Definitions:

is absolutely convergent if converges.

is conditionally convergent if converges but diverges.

Suppose s_n has a mix of positive and negative signed terms.

Ex: s_n = + …

If all of the signs were positive, the sum would be higher because there would not be as much cancellation. If all of the signs were positive, then the series would converge by the p-series test. So it is true that this series also converges because it has the “advantage” of cancellation. This means that the given series converges absolutely (because it still would converge without the negative signs).

If the signs of a series (a) are strictly alternating (not just some random term shave negative signs), (b) decrease in absolute value, and (c) go toward zero, then the series converges (and the error when you stop the series is contained in the first unused term). This is called Leibniz’ Theorem.

Ex: s_n = + … converges (even though the harmonic series diverges!)

Ex: s_n = 12 – 6 + 3 - converges

(Note: s_n = 12 – 6 + 3 - converges because it is a geometric series with | r | < 1, so s_n = )

If s_n converges with its positive and negative signs but it would diverge if all of the signs were the same (such as all positive), then the series converges conditionally (and not absolutely).

Ex: s₁ = converges by Leibniz’ Theorem but

s₂ = diverges by the p-series

So s₁ converges conditionally.