3.3.2 The Integral Test
the integral testis for a series an thatwe cannotevaluate
explicitly
we think the
of series as the union of rectangles with an representing the area of a rectangle ofwidth
one and heightan
visualize the terms of the harmonic series:
11
t
I (7
=
!!!!
itshows the area ofthe shaded columns. it is
bigger than the area under the curve y=b with 12x=5
·Ein S,dx
to continue to infinite the
ifthecolumswere
this
gives us an improper integral we can evaluate
-
(,* (logIx))1P = +
=
the areaunder the curve
diverges to to so the area represented by the columns also diverges to
00
+
example:Enz
Ein ), dx
Eines. Ende
1,*2dx *
=
-
- TE
it 1
=
in
+
the series must converge
Theorem:
letNo be
any natural number. If f(x) is a function which is define and continuous for all xx,No and which obeys
f(x) is
positive for all x =No
f(x) is a
decreasing function
f(n) On for all
= n= No
then Ian converges if (wif(x) dx converges
when the series converges, the truncation error
12an -Ean1= Sf(x) dx for all N =No
the integral testis for a series an thatwe cannotevaluate
explicitly
we think the
of series as the union of rectangles with an representing the area of a rectangle ofwidth
one and heightan
visualize the terms of the harmonic series:
11
t
I (7
=
!!!!
itshows the area ofthe shaded columns. it is
bigger than the area under the curve y=b with 12x=5
·Ein S,dx
to continue to infinite the
ifthecolumswere
this
gives us an improper integral we can evaluate
-
(,* (logIx))1P = +
=
the areaunder the curve
diverges to to so the area represented by the columns also diverges to
00
+
example:Enz
Ein ), dx
Eines. Ende
1,*2dx *
=
-
- TE
it 1
=
in
+
the series must converge
Theorem:
letNo be
any natural number. If f(x) is a function which is define and continuous for all xx,No and which obeys
f(x) is
positive for all x =No
f(x) is a
decreasing function
f(n) On for all
= n= No
then Ian converges if (wif(x) dx converges
when the series converges, the truncation error
12an -Ean1= Sf(x) dx for all N =No
