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Type I -- Infinite Summation


ballon

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observe the pattern in the results, describe it in words. then, formulate a general statement. at this stage, x=1 so a^x=a, so at this stage you do not know yet that the formula is a^x. you only know that it's a.

and then, you'll test for other values of x, and you find that hey it doesn't just approach a, it approaches a^2 when x=2, it approaches a^3 when x=3, etc. so then you conclude that it's a^x :)

for x=2 and a=2, the sum is 2? you sure? maybe you made a calculation mistake?

oh ok thank you very much, so for the stage i am at i just have to right that S(inifinity)= a

i also want to ask you, for the second part, it says that we have to graph between T9(2,x) and x what does that mean i plot what against what? i plot the value of Sn for different values of x? so i find T9(2,2) and the S(infinity) of this is one point?

thank you very much!

No what you would do is calculate each sum to the 9th term of different positive x terms in one graph. For example let x = 1, 2, 3, 4, 5 and calculate the S9 for each one of them. That should be all in one graph. so x axis = x value and y axis = T9 (2, x) value (aka S9 ) So rather than graphing/calculating S1, S2, S3.... S10, you're going straight at S9.

Edited by Excluded
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But why doesn't it work?

Cause when graphing it I can see that it's all messed up, but can I merely exclude x < 0 on the basis that T changes between negative and positive?

Also, really what I need help with is saying in a smart way that x> 0 haha

Edited by Jirashimosu
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After 1 pack of cigarettes and 8 hours I understood this thanks to Desy Glau!!!!! Thank you so much!!!

However another question that I have is. On the second page when they ask us to plot the relation between T9 and x - I am thinking of plotting the 5-6 different values of X that I have chosen on the X - axis and plotting the sum of the(5-6 different) nine terms on y - axis. Is that correct? Because for example when x = 9 Sum of the first nine terms = 460.1 and the expected value is 2^9=512 which is actually met on the 17th term and not the 9th. So what should I plot?

Or is it like different positive values of X on the x-axis and only the different 9th terms on the y-axis? NOT the sum of the first 9 terms but only the 9th term? What is correct? Please help :P

Oh yeah and what type of graph should it be? Should I include trendline or no?

Edited by chicheto93
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After 1 pack of cigarettes and 8 hours I understood this thanks to Desy Glau!!!!! Thank you so much!!!

However another question that I have is. On the second page when they ask us to plot the relation between T9 and x - I am thinking of plotting the 5-6 different values of X that I have chosen on the X - axis and plotting the sum of the(5-6 different) nine terms on y - axis. Is that correct?

you're welcome, and yep that's right :yep:

Because for example when x = 9 Sum of the first nine terms = 460.1 and the expected value is 2^9=512 which is actually met on the 17th term and not the 9th. So what should I plot?

you should plot the T9 (the sum of the first 9 terms)

Or is it like different positive values of X on the x-axis and only the different 9th terms on the y-axis? NOT the sum of the first 9 terms but only the 9th term? What is correct? Please help :P

the sum of the first 9 terms! not just the 9th one.

Oh yeah and what type of graph should it be? Should I include trendline or no?

scatter graph, hmmm yess try to put a nice trendline :yes:

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It says to calculate the sum Sn of the first n terms of the above sequence for 0-10

for this should you get 10 different answers?

yeah, S1, S2, S3, ... , S9 and S10

Do you find what the first term is, then the first term+the second, then first+second+third, and so on?

correct.

Is n what the numerator is raised to?

I just don't understand how to start

n is the thing in the denominator (n!) and also the power in the numerator.

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Hey when it says that the T9(a, x) notation means the sum of the first nine terms

does "first nine terms" refer to

t0, t1, t2, t3, t4, t5, t6, t7, t8, and t9

or just t1, t2, t3, t4, t5, t6, t7, t8, t9

because without t0 = 1 added to the sum....the general statement doesn't really work

OR

could it mean "first nine terms" as in

t0, t1, t2, t3, t4, t5, t6, t7, t8 only?

Edited by Jirashimosu
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Hey when it says that the T9(a, x) notation means the sum of the first nine terms

does "first nine terms" refer to

t0, t1, t2, t3, t4, t5, t6, t7, t8, and t9

or just t1, t2, t3, t4, t5, t6, t7, t8, t9

because without t0 = 1 added to the sum....the general statement doesn't really work

OR

could it mean "first nine terms" as in

t0, t1, t2, t3, t4, t5, t6, t7, t8 only?

The way I see it, it's the "first nine terms" as in 0 < n < 9. I mean, the sum of the first 10 terms would probably be from 0 to 10, right? Though don't quote me on it :P

Now for my own question (I'm not sure if it's been addressed yet, so I apologize if it has): why do they specifically ask us to consider x=1,a=2 and x=1,a=3 and THEN ask us to consider various other sequences where x=1? Is there any purpose to it or is it just so that we get the hang of things? Maybe I've been spending too much time in ToK...

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How do i test the validity of the general statement , and discuss limitations and explain all this (second page) and the general statement for the first part is a^x but how do i get the 2nd part's general statement?

I've got the same question! Can anyone help?

And also can anyone give me hints as to how to provide an informal justification for the general statement?

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To test the validity of your general statement you should give different values for a and x, and then represent them in the way you have previously done. Don't forget to calculate the values for the first nine terms only.

For scope and/or limitations consider real numbers, irrational numbers, negative integers, and etc to see whether they would work for your general statement.

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