• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# 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= 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 As n  +, Sn  +2

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

tn= 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 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). 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.

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 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 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= 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 As n → +∞, Sn → +6

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

tn = 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 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 = 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 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 = 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 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 = 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

Conclusion

tn = 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 As n → +∞, Sn → + 11

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

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Extended Essay- Math

ï¿½ï¿½[ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½}ï¿½ï¿½ï¿½ï¿½Ëï¿½ \$ï¿½ï¿½FYï¿½"8oï¿½Tï¿½>ï¿½-ï¿½yï¿½"oï¿½ï¿½ï¿½]Gï¿½ï¿½ï¿½ï¿½>ï¿½?ï¿½ï¿½ï¿½...'Ð£,ï¿½ï¿½7ï¿½*ï¿½S?ï¿½-ï¿½<ï¿½ï¿½7ï¿½ï¿½(r)ï¿½ï¿½fï¿½?ï¿½gï¿½rï¿½ï¿½ï¿½ï¿½ï¿½?ï¿½Q-ï¿½ ï¿½ï¿½ ï¿½)ï¿½"ï¿½K~-ï¿½ï¿½ï¿½WQï¿½ï¿½qï¿½ï¿½ï¿½ï¿½9c?ï¿½padï¿½ï¿½(ï¿½?ï¿½ ï¿½Êï¿½ï¿½ï¿½ï¿½%ï¿½?ï¿½ï¿½+ï¿½ï¿½Yï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½?ï¿½ï¿½Oï¿½eï¿½ï¿½Cï¿½ï¿½ï¿½Aï¿½ cï¿½ ï¿½ßï¿½ï¿½Fï¿½{ï¿½ï¿½ï¿½ï¿½ï¿½Gï¿½Cï¿½ï¿½ Xï¿½\ï¿½Y'ï¿½ 2ï¿½!ï¿½ï¿½ ï¿½...1ï¿½oï¿½ï¿½#|=ï¿½ï¿½ï¿½?ï¿½n#ï¿½ï¿½ï¿½qï¿½ï¿½,gï¿½.ï¿½,"ï¿½...gï¿½ï¿½PÂï¿½Aï¿½D*ï¿½ï¿½3/4-ï¿½ï¿½uï¿½7ï¿½ï¿½8ï¿½ï¿½-3ï¿½- ï¿½Iï¿½Bï¿½ï¿½ï¿½pï¿½ï¿½ï¿½e?ï¿½ï¿½^ï¿½ï¿½ï¿½ï¿½z-ï¿½ï¿½ï¿½-Qï¿½ï¿½ï¿½...ï¿½ï¿½ï¿½ï¿½'L1/4I(c)ï¿½\$iï¿½ï¿½ï¿½mï¿½ï¿½ï¿½Mï¿½Xkï¿½ï¿½ï¿½ï¿½/ï¿½ï¿½s] (ï¿½ï¿½ï¿½ï¿½1ï¿½9ï¿½ï¿½tï¿½T"\ï¿½wï¿½ï¿½uwg(c)ï¿½+ï¿½pï¿½l._ï¿½ï¿½ï¿½Iï¿½Uï¿½...ï¿½Jï¿½ï¿½3/4(ï¿½ï¿½ï¿½ï¿½ï¿½^ï¿½(c)ï¿½Wï¿½ï¿½ï¿½:ï¿½ï¿½ï¿½ï¿½ï¿½l?ï¿½7ï¿½-ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ ï¿½ï¿½ï¿½ ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ kHï¿½Èï¿½!aï¿½ï¿½Tn`ï¿½NXï¿½ï¿½Íp9<Pï¿½(r)ï¿½ÑÕU"ï¿½V-ï¿½ï¿½ ï¿½phï¿½ï¿½ï¿½A9-A ï¿½=Aï¿½@ @hï¿½{ï¿½ï¿½1/2ï¿½ï¿½ -,ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Mrï¿½?ï¿½.ï¿½?ï¿½ï¿½ "ï¿½"aï¿½ï¿½ï¿½ï¿½1/4ï¿½5ï¿½ï¿½}ï¿½P ï¿½ @[email protected]@[email protected]@[email protected]@[email protected]@P Âï¿½ï¿½ï¿½ï¿½;ï¿½ï¿½3/4ï¿½ï¿½ï¿½[ï¿½Fï¿½ï¿½_ï¿½Vï¿½Cï¿½ ;Aï¿½|aï¿½ï¿½dï¿½"|Uï¿½1/2ï¿½1jï¿½7ï¿½lï¿½ï¿½Wï¿½^ï¿½ï¿½ï¿½ #ï¿½Iï¿½2Ö³Cï¿½ï¿½_ ï¿½Oï¿½ï¿½ï¿½<#ï¿½l3/4*ï¿½"ï¿½ï¿½~ï¿½S^}ï¿½ï¿½- ï¿½4 Vï¿½_ï¿½*|xï¿½ï¿½ï¿½ï¿½kj:Ö£ï¿½h~ï¿½C"ï¿½ H'ï¿½ï¿½ï¿½Qï¿½-ï¿½0kï¿½X~ï¿½ï¿½mï¿½ï¿½Oï¿½zï¿½ï¿½ h>1ï¿½ï¿½}3Eï¿½ï¿½ï¿½Å¦iï¿½ï¿½'ï¿½ï¿½\$ï¿½ï¿½CgBï¿½:ñ|_ï¿½4ï¿½nï¿½ï¿½ï¿½i(c)...ï¿½P "ï¿½/ï¿½ï¿½ï¿½ï¿½9h|e x"ï¿½ï¿½?| 'ï¿½Oï¿½-"ï¿½> ï¿½ ï¿½ï¿½hï¿½ï¿½ï¿½5ï¿½^ï¿½ï¿½(r)ï¿½ï¿½Zï¿½I'ï¿½uA4V(r)ï¿½ï¿½e""Wï¿½ï¿½+"p5ï¿½ï¿½ï¿½+ï¿½cswï¿½ï¿½NYVSï¿½ï¿½ï¿½ln2Uï¿½ï¿½ï¿½&ï¿½ZxjPJNï¿½ï¿½W'%cï¿½ï¿½<Aï¿½ï¿½...ï¿½dXï¿½>ï¿½Uï¿½ï¿½cï¿½AVï¿½ï¿½ï¿½ï¿½ï¿½ï¿½'pï¿½j'n<ï¿½Ò"ï¿½ï¿½ï¿½ï¿½Jï¿½xï¿½ï¿½3/44ï¿½ï¿½ï¿½ï¿½-ï¿½ï¿½...jï¿½ï¿½ï¿½ï¿½Uï¿½Kï¿½ï¿½oï¿½h6ï¿½ï¿½ï¿½"<ï¿½ï¿½mï¿½(r).-ï¿½ï¿½Gko-ï¿½ï¿½h"-~ï¿½ï¿½ï¿½ ,ï¿½ï¿½' ï¿½*ï¿½0YM wdY...L&ï¿½hï¿½Xï¿½U\|ï¿½+E^sï¿½Pï¿½(c);ï¿½q"\ï¿½woï¿½1/4-ï¿½cqY>pï¿½ï¿½ï¿½F:(r)ï¿½3,+ï¿½jJï¿½GJï¿½,*ï¿½1/4ï¿½ï¿½Ü¥ï¿½F.Rï¿½Jï¿½%ï¿½ ï¿½/ï¿½ï¿½i/Ú£ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½_?fï¿½Ù¶ï¿½ï¿½ZGï¿½ _Cï¿½-ï¿½ï¿½|Bï¿½ï¿½ï¿½;ï¿½Z <]%ï¿½ï¿½ï¿½ï¿½Z.ï¿½ï¿½7)ï¿½\ï¿½ï¿½ï¿½[eï¿½Apï¿½} ï¿½'ï¿½pgpï¿½Sï¿½|Qï¿½|eW0ï¿½e8|N(tm)arï¿½ï¿½/(c)

2. ## Math Studies I.A

\$19,100 (2007 est.) \$18,500 (2006 est.) note: data are in 2008 US dollars Belarus \$11,800 (2008 est.) \$10,700 (2007 est.) \$9,900 (2006 est.) note: data are in 2008 US dollars Belgium \$37,500 (2008 est.) \$37,100 (2007 est.) \$36,200 (2006 est.)

1. ## Math IA - Logan's Logo

have: a=-2.2 b= c=3.4 d=-1.3 y=-3.15sin(1.01x+3.0)+0.35 Combining the determined values of a, b, c, and d, we now have a complete sine function to fit the data with: This then gives us the following graph: However, we can see, in the blue circled area, places where the curve doesn't fit the data well.

2. ## Math Studies - IA

These two numbers can then be compared. Whenever the calculated value is below 1, US wins, and when it is above 1, Europe wins. A value of 1 represents a tie because for example a Ryder Cup final score of 14 - 14 is ()

1. ## Infinite Summation Internal Assessment ...

Here is another example of the sequence taking the same concept where , a x n t Sn 2 7 0 1.000000 1.000000 1 4.852030 5.852030 2 11.771099 17.623129 3 19.037909 36.661038 4 23.093128 59.754166 5 22.409711 82.163878 6 18.122099 100.285977 7 12.561282 112.847259 8 7.618465 120.465724 9 4.107225 124.572949

2. ## SL Type 1 PF - Infinite Summation - A general statement has been reached, ...

Figure 4: Values of when Along with the data values of , the exponential graph of is plotted for comparative analysis. Observing the values suggests that the exponential graph of matches the values of , which allows a generalized statement: In order to reach a conclusion, values of for , , and will be analyzed.

1. ## This assignments purpose is to investigate how translation and enlargement of data affects statistical ...

Height (cm) Frequency Cumulative Frequency 130-139 17 17 140-149 14 31 150-159 5 36 160-169 9 45 170-179 15 60 The graph below would represents

2. ## Infinite Summation- The Aim of this task is to investigate the sum of infinite ...

Now a general sequence where x=1, will be considered. Taking all data collected into account a relation between a and ? has shown. So as the two sets of results show a=2 or a=3, the total of sum won?t exceed 2 or 3, when x=1. To prof this suggestion right, two more examples will be made, again when x=1 • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 