The_integral_Test.ppt

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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example
Test the series for convergence or divergence.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
Example
Remark
Test the series for convergence or divergence.
sequence of positive terms.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
Example
Test the series for convergence or divergence.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
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
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
REMARK
REMARK
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.
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
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
Special Series
Example

1. Geometric Series
2. Harmonic Series
3. Telescoping Series
4. p-series
5. Alternatingp-series

Harmonic Series
is the series convergent?
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Memorize
Special Series

1. Geometric Series
2. Harmonic Series
3. Telescoping Series
4. p-series
5. Alternatingp-series

Example
For what values of p is the series convergent?
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
Example
P Series
For what values of p is the series convergent?
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example
P Series
For what values of p is the series convergent?
Example
Example
Test the series for convergence or divergence.
Test the series for convergence or divergence.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
FINAL-081
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM (Integral Test)
a continuous, positive, decreasing function on
1, inf)
Convergent
Convergent
Divergent
Dinvergent
REMARK
REMARK
We should not infer from the Integral Test that
the sum of the series is equal to the value of
the integral. In fact,
Integral Test just test if convergent or
divergent. But if it is convergent what is the
sum??
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Bounds for the Remainder in the Integral Test
Convergent by integral test
Error (how good)
good approximation
Example
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK
We can estimate the sum
ESTIMATING THE SUM OF A SERIES

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

1000 1.643934566681562
11000 1.644843161889427
21000 1.644886448934383
61000 1.644901809303995
71000 1.644919982440396
81000 1.644921721245446
91000 1.644923077897639
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK
We can estimate the sum
ESTIMATING THE SUM OF A SERIES
Example
Estimate the sum
How accurate is this estimation?
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example
Estimate the sum
How accurate is this estimation?
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Sec 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS
Facts about (Harmonic Seris)
1)The harmonic series diverges, but very slowly.
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the sum of the first million terms is less than
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the sum of the first billion terms is less than
2) If we delete from the harmonic series all
terms having the digit 9 in the denominator.
The resulting series is convergent.
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-112
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THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102
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