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II - Convergence: Discrete variables
S5 = (1 - 1/2) + (1/3 - 1/4) + 1/5.
It thus converges to a limit s ;:::: o.
The sums of even order
S2n = S2n-l - 1/2n
also converge to s since 1/2n tends to O. It is now clear that the series (1)
converges and has sum s.
These arguments, like the following theorem due to Leibniz, extend immediately to alternating series, those of the form
Ul - U2 + U3 - ... ,
where the Un are all positive.
Theorem 6. Every alternating series whose terms tend to 0 while decreasing
in absolute value, is convergent.
Now, the partial sums
decrease and are positive since
In consequence S2n+1 tends to a limit s ;:::: 0, which is also the limit of S2n
since U2n+l tends to 0, whence the convergence of the series.
Note that while the S2n+l decrease, the S2n, on the contrary, increase; so
(13.2)
for all p and q. For p = q = n, this relation may be written
or
We deduce that
(13.3)
for any r. In other words, the error committed in replacing the total sum by
a partial sum is, in absolute value, smaller than the first term neglected. Nor
is it much smaller: the relation