An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 43- 44.

General theorems concerning series.

73

Therefore in general the interval of convergence (convergency)
of the series is given by the condition
Lim +1 x < 1, or x < Lim a (n = ).
an * an+i
II. The sum (difference) of two convergent series is itself a convergent series, whose terms consist of the sum (difference) of the
terms of both.
If f(x) -= a -+- a, x + a., x2+ - a n-. n -_ + - n,
< (x) = bo + bI x + bX2 +. -1 xn-1 + B'n,
be such, that a determinate n can be chosen, so that R as well as I'
may become less than any arbitrarily small number, we have
/(x) + (x)= ( (at, )+ (a+ ib) X+ (an-i + b-1) n-' + +B'.
Now since we have Lim (, +- R'n) = 0, for all values of x for which
both series converge, we obtain for the algebraic sum the infinite series
f (x) + { (x) = ao ~ b0 + (a, + b1) x + (a2 + b.) x2 +
Still more generally if the series converge respectively for x and x', we have
f(x) + 9 (x') = (ao + bo) + (a x + b1 x') + (a2 x + b2 X2) +...
III. An infinite series, whose terms have different signs for some
value of x, converges, if the limit of the sum of the positive terms
be finite and also the limit of the sum of the negative terms be finite.
For by Theorem II such a series expresses the difference of the
values of two convergent series. When this is the case, the series
consisting of the same terms taken all with like sign converges and
we shall see that it has the same limiting value even when the order
of its terms is changed: such a series is said to be absolutely (unconditionally) convergent. But a series whose terms have different signs may converge without the sum of the positive and of the
negative terms separately having finite limits, it is then said to be
semiconvergent (conditionally convergent). A series converges
unconditionally when the absolute value of the quotient of a term
by the preceding one is less than unity for all values from some
determinate n on to n ==-oo. For then, even when all the terms are
written with the same sign, the series fulfils the condition of convergence proved sufficient in Theorem I.
It is thus seen: that a conditional convergence can only arise by
the ratio of a term to the precedinlg one being less in amount than
unity, but becoming unity for n =oo; and hence follows further:
If a series of powers of x converges only conditionally for a determinate value X then it converges absolutely for every numerically smaller
value of x; while it diverges for a greater value.
For, for a smaller value the quotient remains less than one, for