Calculus: Analytic Geometry and Calculus, with Vectors

5.6 Sequences, series, and decimals 341

(1646-1716) and others before Euler. A given series ul + u2 + u3 + is
assigned the value Y by this method if the series

(2) ul + u2r + u3r2 +...

converges when 0 < r < 1 to the values f(r) of a function for which

(3) lim f (r) = V.

r-+1-

Use these ideas to find the Abel value of the series 1 - 1 + 1 - 1 +.

Hint: The series 1 - r + r2 - r3 + is a geometric series whose ratio is

-r, and the series converges to1/(1 + r) when Irl < 1 Remark: Our mathe-

matical notations would be more sensible but less brief if we were accustomed to

w citing

(4) s =C{ul+u2+u3+ ...}

to abbreviate the statement that the series in braces is assigned the value s by

the method of convergence and to writing

(5)

to abbreviate the statement that the series in braces is assigned the value V by

the method of Abel. This more elaborate notation can show just what we are

doing when we adopt the convenient but absurd old idea that a conglomeration

of numbers and plus signs "is" a number or "represents" a number if and only

if it converges to the number. An intelligible theory of series requires a suitable

mixture of broad ideas of Euler and narrow ideas usually promoted by elementary

books of the nineteenth and twentieth centuries.

7 Each sequence sl, Si, s3, of numbers determines its sequence M1, M2,

M3, - of arithmetic means defined by the formulas

(1) Mn =

Jl + S2 + .. S.

= sk (n = 1,2,3...

If

k=1

(2) lim sn = s,

n--

so that s is near s whenever n is large, we can feel that Mn shouldalso be near

s whenever n is large and hence that

(3) lim Mn = J.

n-.

Prove that (2) implies (3). Solution: Let E > 0. Choose an integer N such that

Isn - sI < e/2 whenever n > N. Then, when n > N,

(4) Mn - s =(sl - s) + (s2 - s) +

... + (sn - s)

(sk - s)

n k-l

and hence

n

(5) IMn-sl5 ' Isk-sl+n Isk-sl;5 +n 2<n+ 2,

k=1 k- +1 k= 1