(4) Integral Test
We need to define the monotonic sequences before the integral test which as the following:
Monotonic Sequences:
for all n, it is is increasing if A sequence
for all n. A monotonic sequence decreasing if
is one which is either increasing or decreasing.
Example:
is monotonic (decreasing). (1)
is monotonic (increasing). (2)
Remark:
The increasing or decreasing series is applying the similarly conditions above .
Integral Test:
The integral test is a powerful way to study the convergence and approximate value of a monotonically decreasing series.
are identified with a monotonically If the terms of the series then the series is so that decreasing function
bounded above and below by the value of the integral.
Thus, the convergence or divergence of the series is the same as the convergence or divergence of the integral.
Remark:
is a continuous, positive, decreasing where Let and the integral .Then the series function of x for all
both converge or both diverge.
In other words,
be a decreasing series of positive terms, and let Let
be a continuous function obtained on the
, so we have interval
Converges. finite, then (1) If
diverges. infinite, then (2) If
Example:
Use the Integral Test to investigate the convergence or divergence of the following series:
Solution:
, and , then If
is decreasing series .
finite
converges .
Example:
Use the Integral Test to investigate the convergence or divergence of the following series:
Solution:
, and , then If
is decreasing series .
infinite
diverges .
Exercise:
Use the Integral Test to investigate the convergence or divergence of the following series:
.
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