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The purpose of this paper is to investigate an infinite summation patter where Ln(a) is a constant and the coefficient of x is an increasing factor to Ln(a).

Extracts from this document...

Introduction

Infinite Summation

Math IA

        The purpose of this paper is to investigate an infinite summation patter where Ln(a) is a constant and the coefficient of x is an increasing factor to Ln(a).

Consider the following sequence of terms where x=1 and a=2 under the terms that 0≤n≤10:

tn=image00.png

n

t(n)

S(n)

0.000000

1

1

1.000000

0.69314718

1.693147

2.000000

0.24022651

1.933373

3.000000

0.05550411

1.988877

4.000000

0.00961813

1.998495

5.000000

0.00133336

1.999829

6.000000

0.00015404

1.999983

7.000000

1.5253E-05

1.999998

8.000000

1.3215E-06

1.999999

9.000000

1.0178E-07

1.999999

10.000000

7.0549E-09

2

image01.png

        As n  +, Sn  +2

Consider the following sequence of terms where x=1 and a=3:

tn=image12.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

1.098612

2.098612

2.000000

0.603474

2.702087

3.000000

0.220995

2.923082

4.000000

0.060697

2.983779

5.000000

0.013336

2.997115

6.000000

0.002442

2.999557

7.000000

0.000383

2.999940

8.000000

0.000053

2.999993

9.000000

0.000006

2.999999

10.000000

0.000001

3.000000

image23.png

As n → +∞, Sn → +3

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive three. Sn approaches a horizontal asymptote when y=3. There is a y-intercept at (0,1).

image32.png

As n → +∞, Sn → +4

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive four. Sn approaches a horizontal asymptote when y=4.

...read more.

Middle

5.666868

3.000000

0.277521

5.944389

4.000000

0.048091

5.992480

5.000000

0.006667

5.999146

6.000000

0.000770

5.999917

7.000000

0.000076

5.999993

8.000000

0.000007

5.999999

9.000000

0.000001

6.000000

image05.png

As n → +∞, Sn → +6

Let a=2 and calculate various positive values for x:

x

t(n)

S(n)

0.0

1.000000

1.000000

1.0

0.693147

1.693147

2.0

0.480453

2.173600

3.0

0.166512

2.340113

4.0

0.038473

2.378585

5.0

0.006667

2.385252

        In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.

Let a=3 then calculate for various positive values of x:

x

t(n)

S(n)

0.0

1.000000

1.000000

1.0

1.098612

2.098612

2.0

1.206949

3.305561

3.0

0.662984

3.968546

4.0

0.242788

4.211333

5.0

0.066682

4.278016

image06.png

In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.

Evidence:

tn=image07.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

2.772589

3.772589

2.000000

0.960906

4.733495

3.000000

0.222016

4.955511

4.000000

0.038473

4.993984

5.000000

0.005333

4.999317

6.000000

0.000616

4.999933

7.000000

0.000061

4.999994

8.000000

0.000005

5.000000

9.000000

0.000000

5.000000

image08.png

As n → +∞, Sn → +6

Sn approaches a horizontal asymptote when y=6. There is a y-intercept at (0,1).

tn = image09.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

4.158883

5.158883

2.000000

1.441359

6.600242

3.000000

0.333025

6.933267

4.000000

0.057709

6.990976

5.000000

0.008000

6.998976

6.000000

0.000924

6.999900

7.000000

0.000092

6.999991

8.000000

0.000008

6.999999

9.000000

0.000001

7.000000

image10.png

As n → +∞, Sn → +7

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive seven. Sn approaches a horizontal asymptote when y=7. There is a y-intercept at (0,1).

tn = image11.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

4.852030

5.852030

2.000000

1.681586

7.533616

3.000000

0.388529

7.922145

4.000000

0.067327

7.989471

5.000000

0.009333

7.998805

6.000000

0.001078

7.999883

7.000000

0.000107

7.999990

8.000000

0.000009

7.999999

9.000000

0.000001

8.000000

image13.png

As n → +∞, Sn → +8

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive eight. Sn approaches a horizontal asymptote when y=8. There is a y-intercept at (0,1).

tn = image14.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

5.545177

6.545177

2.000000

1.921812

8.466990

3.000000

0.444033

8.911022

4.000000

0.076945

8.987967

5.000000

0.010667

8.998634

6.000000

0.001232

8.999867

7.000000

0.000122

8.999989

8.000000

0.000011

8.999999

9.000000

0.000001

9.000000

image15.png

As n → +∞, Sn → + 9

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive nine. Sn approaches a horizontal asymptote when y=9. There is a y-intercept at (0,1).

tn = image16.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

6.238325

7.238325

2.000000

2.162039

9.400363

3.000000

0.499537

9.899900

4.000000

0.086563

9.986463

5.000000

0.012000

9.998464

6.000000

0.001386

9.999850

7.000000

0.000137

9.999987

8.000000

0.000012

9.999999

9.000000

0.000001

10.000000

10.000000

0.000000

10.000000

...read more.

Conclusion

tn = image19.png

n

t(n)

S(n)

0.000000

1.000000

1.000000

1.000000

5.493061

6.493061

2.000000

3.017372

9.510434

3.000000

1.104974

10.615408

4.000000

0.303485

10.918893

5.000000

0.066682

10.985575

6.000000

0.012210

10.997785

7.000000

0.001916

10.999701

8.000000

0.000263

10.999964

9.000000

0.000032

10.999996

image20.png

As n → +∞, Sn → + 11

There is a horizontal asymptote as n approaches positive infinite (∞).

image21.png

tn= image22.png

n

t(n)

S(n)

0.0

1.000000

1.000000

1.0

7.690286

8.690286

2.0

4.224321

12.914607

3.0

1.546964

14.461571

4.0

0.424878

14.886450

5.0

0.093355

14.979805

6.0

0.017094

14.996898

7.0

0.002683

14.999581

8.0

0.000368

14.999950

9.0

0.000045

14.999995

image24.png

As n → +∞, Sn → + 15

There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive fifteen. Sn approaches a horizontal asymptote when y=15. There is a y-intercept at (0,1).    

tn= image25.png

n

t(n)

S(n)

0.0

1.000000

1.000000

1.0

8.788898

9.788898

2.0

4.827796

14.616694

3.0

1.767959

16.384653

4.0

0.485575

16.870228

5.0

0.106692

16.976920

6.0

0.019535

16.996455

7.0

0.003066

16.999521

8.0

0.000421

16.999942

9.0

0.000051

16.999994

image26.png

As n → +∞, Sn → + 17

There is a horizontal asymptote as n approaches positive infinite (∞). Sn approaches a horizontal asymptote when y=17. There is a y-intercept at (0,1).    

Checks with various numbers:

tn =image27.png

n

t(n)

S(n)

0.0

1.000000

1.000000

1.0

48.283137

49.283137

2.0

38.854356

88.137493

3.0

20.844558

108.982051

4.0

8.387005

117.369057

5.0

2.699673

120.068729

6.0

0.724159

120.792889

7.0

0.166498

120.959387

8.0

0.033496

120.992883

9.0

0.005990

120.998873

10.000000

0.000964

120.999837

image28.png

tn = image29.png

n

t(n)

S(n)

0.0

1.000000

1.000000

1.0

13.961881

14.961881

2.0

32.489020

47.450901

3.0

50.400871

97.851772

4.0

58.640914

156.492686

5.0

54.582497

211.075184

6.0

42.337463

253.412647

7.0

28.148125

281.560772

8.0

16.375032

297.935804

9.0

8.467639

306.403443

10.000000

3.940806

310.344249

n

t(n)

S(n)

0.0

1.000000

1.000000

1.0

179.175947

180.175947

2.0

160.520100

340.696047

3.0

95.871136

436.567183

4.0

42.944504

479.511687

5.0

15.389244

494.900931

6.0

4.595637

499.496569

7.0

1.176325

500.672894

8.0

0.263461

500.936356

9.0

0.052451

500.988807

10.000000

0.009398

500.998205

image30.png

tn = image31.png

Conclusion:

The limitation to this general statement is that the scope of the evidence is limited to only a few combinations of x and a .  

The continuous observation of various graphs with different values used in place of a and x provide evidence for the general statement.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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