For what values of p does \(\int_{1}^{\infty} x^{-p} d x\) converge?
Step 1 of 6
Definition ;
The partial sum of the series
is given by
=
+
+
+..............+
. If the sequence of these partial sums {Sn} converges to L,
then the sum of the series converges to L. If {Sn} diverges, then the sum of the series diverges.
=
is convergent . Conversely , a series is divergent if the sequence of partial sums is divergent. If
and
are convergent series , then
+
) and
-
) are convergent . If C
, then
is convergent series.
NOTE : The terms grow without bound , so the sequence does not converge.
Integral test ; The series can be compared to an integral to establish convergence or divergence.
Let f(n) = be a positive monotone decreasing function . If
f(x) dx =
f(x) dx <
,
then the series converges. But if the integral diverges, then the series does so as well.
