Integral Test
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Conditions of Integral test
Assume the series a𝖓
can be represented as a function f(x)
. There are a few limitations for it to use the Integral test:
f(x)
MUST BE continuous.f(x) > 0
. It MUST BE a positive function.f'(x) < 0
. It's MUST BE decreasing.
Using Integral test
Example
Understanding Integral test
The Integral test
has introduced the idea of calculating the total area under the function:
The series has step of 1, which means
Δx = 1
We can sum the areas (which equals the series itself):
But when we are to INTEGRATE the function area under the function:
The
dx
is infinitely small rather than a fixed numberΔx = 1
.As result, the INTEGRAL is almost always greater than the SERIES AREAS.
Notice: DO NOT use the
Integral Test
to EVALUATE series, because in general they are NOT equal.
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