Case II. divergent.

r = 1. The series can not be said to be convergent or For consider the series (3). Then

Then r =

1, no matter what p is. But when p> 1, (3) converges (2653); and when p < 1, (3) diverges (3656, Ex. 2). That is, r may equal 1 both for a convergent and for a divergent series.

Un+1

NOTE. The student should note that the theorem requires that the limit of the ratio

un series,

is always less than 1, in case the series is convergent. Thus, in case of the harmonic

'n +1

Here the ratio

is less than 1 for all values of n, yet the series is divergent (§655); un but the limit of the ratio is not less than 1 but equal to it.

Therefore the series is convergent for all finite values of x.

is convergent if n is greater than 2, divergent if n is less than

or equal to 2.

14. Suppose that in the series u+u, +"2+ "3 +... ... each term is less than the preceding; then show that this series and the series

1

convergent or both divergent.

15

SERIES WITH POSITIVE AND NEGATIVE TERMS

are both

659. Alternating Series.-THEOREM. Suppose that the terms of the given series are alternately positive and negative,

and that each term is less than or equal to the one which precedes it,

lim

and let nun = 0; then the series is convergent.

Throughout the steps of the proof which is to follow, consider as an example the series

Outline of the Plan of the Proof of the Theorem

and plot the points 8,, 82, 83,

that the points 81, 8, 857

S2r+1)

move to the left, but

to the right but never advance as far to the right as s1; hence according to the same theorem they approach a limit, U2;

2

but, according to the third part of the hypothesis of the theorem,

or let us say U. That is 8, approaches a limit U, continually oscillating from one side of its limit to the other.

It is now required to establish analytically the facts on which the plan of the proof rests. It is required to prove:

= (μ — u2) + · · · + (Ugr_4 — U2r-3) + (U2r-2 — U2r-1) = 8 2r_2 + (U2r_2 – ·U2r_1);

where the parentheses according to the second part of the hypothesis are all positive (or zero). Hence, the values of 821 Ꭶ

s,

69

continually increase, and the values of 8, 8, 8, . . . continually decrease.

or to four places, the value of the series is .4055.

The Limit of Error in an Alternating Series

In calculating the value of an infinite series, it is an important matter to know that the value of the series is correct to a given number of decimal places, say to four places.

In order to determine the value of an alternating series correct to the fourth decimal place, it is not sufficient to know that the series is convergent, and that therefore enough terms can be taken so that their sum, 8, will differ from the limit U of the series by less than .0001, since the series might converge so slowly that it would be necessary to take n = 100,000 or greater, so that it would be practically impossible to compute so many terms.

RULE.-The sum of the first n terms of an alternating series, (5), 2659, 8, differ from U, the value of the series, by less than the (n+1)th term. Hence we can stop computing terms as soon as a term is reached which is numerically less than the proposed limit of error.

8n+1

The proof of this rule follows from the discussion connected with the figure on page 639. is determined from s1 by adding ± a quantity which is greater than the distance from s, to U. But un is the (n+1)th term of the series. This proves the rule.

6. Compute correct to three decimal places the value of

7. Compute the value of the series in Example 3 correct to four decimal places.

be a convergent series of positive and negative terms.

where r is an integer which is a constant or varies with n.

Proof. Let $n = U ̧ + U1 +

and plot the points 8,, 8, 831

1

Now, when we say that

the u series is convergent we mean that s, approaches a limit "; that is, that there is a point U about which the values of s, arrange themselves as n increases. In all the series thus far discussed, always came nearer to U as n increased; this is not necessarily

required by the hypothesis of this theorem. Thus s, may be farther away from U than s But the hypothesis of this theorem does require that ultimately s, may be made to differ from U by as small