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