Upload
samson-howard
View
219
Download
1
Tags:
Embed Size (px)
Citation preview
THE INTEGRAL TEST AND ESTIMATES OF SUMS
Test the series for convergence or divergence.
12
1
n n
Example:
dxx
12
1
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf
nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
Test the series for convergence or divergence.
12 1
1
n n
Example:
sequence of positive terms.
Remark:
1nna
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf
nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
Test the series for convergence or divergence.
1
ln
n n
n
Example:
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series
12)5.2(
1
n n
REMARK:
3 2)5.2(x
dx
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
Also, it is not necessary that f(x) be always decreasing. What is important is that f(x) be ultimately decreasing, that is, decreasing for larger than some number N.
REMARK:When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series
12)5.2(
1
n n
REMARK:
3 2)5.2(x
dx
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
is the series convergent?
1
1
n n
Example: Harmonic Series
Special Series:
1) Geometric Series
2) Harmonic Series
3) Telescoping Series
4) p-series
5) Alternatingp-series
1
1
n
nar
1
1
nn
11)(
nnn bb
1
1
npn
THE INTEGRAL TEST AND ESTIMATES OF SUMS
For what values of p is the series convergent?
1
1
npn
Example:Special Series:1) Geometric Series
2) Harmonic Series
3) Telescoping Series
4) p-series
5) Alternatingp-series
1
1
n
nar
1
1
nn
11)(
nnn bb
1
1
npn
Memorize:
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
For what values of p is the series convergent?
1
1
npn
Example:P Series:
1
11
1 pdivg
pconvg
nnp
THE INTEGRAL TEST AND ESTIMATES OF SUMS
For what values of p is the series convergent?
1
1
npn
Example:P Series:
1
11
1 pdivg
pconvg
nnp
13
1
n n
Example:
13/1
1
n n
Example:Test the series for convergence or divergence.
Test the series for convergence or divergence.
THE INTEGRAL TEST AND ESTIMATES OF SUMS
FINAL-081
THE INTEGRAL TEST AND ESTIMATES OF SUMS
a continuous, positive, decreasing function on [1, inf)
Convergent
1nna
1)( dxxf
THEOREM: (Integral Test)
)(xf nanf )(
Convergent
Divergent
1nna
1)( dxxf Dinvergent
Integral Test just test if convergent or divergent. But if it is convergent what is the sum??
REMARK:We should not infer from the Integral Test that the sum of the series is equal tothe value of the integral. In fact,
REMARK:
12
1
n n1
11 2
dxx6
2
THE INTEGRAL TEST AND ESTIMATES OF SUMS
Convergent by integral test
1mma
Bounds for the Remainder in the Integral Test
1m
maS
n
mmn as
1
nn sSR
nnndxxfRdxxf )()(
1
1nmmn aR
good approximationError (how good)
68482261.644934066
1 2
12
n n
6439345666.10001 SExample:
THE INTEGRAL TEST AND ESTIMATES OF SUMS
We can estimate the sumREMARK:6
1 2
12
n n
ESTIMATING THE SUM OF A SERIES 68482261.644934066
1 2
12
n n
1 1.000000000000000
2 1.250000000000000
3 1.361111111111111
4 1.423611111111111
5 1.463611111111111
10 1.549767731166541
20 1.596163243913023
40 1.620243963006935
50 1.625132733621529
nsn22222 5
1
4
1
3
1
2
1
1
15 s
1000 1.643934566681562
11000 1.644843161889427
21000 1.644886448934383
61000 1.644901809303995
71000 1.644919982440396
81000 1.644921721245446
91000 1.644923077897639
nsn
THE INTEGRAL TEST AND ESTIMATES OF SUMS
ESTIMATING THE SUM OF A SERIES
21
sum partial
211
nn
thn
nn
n aaaaaa
sum partial
211
thn
nn
n aaaa
ns
13
1
n n
Example:Estimate the sum
How accurate is this estimation?
We can estimate the sumREMARK:6
1 2
12
n n
nnndxxfRdxxf )()(
1
10SS
n nx
dx23 2
1
200
1
242
110 SS
005.0100041.0 SS
THE INTEGRAL TEST AND ESTIMATES OF SUMS
13
1
n n
Example:Estimate the sum
How accurate is this estimation? 10SS
n nx
dx23 2
1
200
1
242
110 SS
005.0100041.0 SS
10005.0100041.0 SSS
2025.12016.1 S1975.1005.01975.10041.0 S
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
Facts about: (Harmonic Seris)
1
1
n n
1)The harmonic series diverges, but very slowly.
the sum of the first million terms is less than 15the sum of the first billion terms is less than 22
2) If we delete from the harmonic series all terms having the digit 9 in the denominator. The resulting series is convergent.
THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-112
THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102