- #26

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ya agreed.

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- Thread starter dilan
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- #26

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ya agreed.

- #27

arildno

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What happens to the value of Sn as N grows bigger and bigger?

- #28

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What does |x|<1 this mean? i mean by using |x|

- #29

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The value gets smaller and smaller right?

- #30

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Either positive or negative values right?

- #31

arildno

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So the absolute value of "2" is 2, and the absolute value of "-2" is also 2.

The absolute value is simply the distance of a number on the number line from the origin.

- #32

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So because of the absolute value are we getting positive values for X?

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- #33

arildno

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The term [itex]x^{N}[/itex] will approach zero as N towards to infinity if |x|<1.

Do you agree to that?

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- #35

arildno

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If you multiply a positive number that is less than one with itself, will the product be less than or greater than the number itself?

- #36

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Well it say the number = y

and |y|<1

then the product will be less than the number right?

and |y|<1

then the product will be less than the number right?

- #37

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I mean when you multiply it by itself.

- #38

arildno

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So when you multiply itself with itself N times, where N is some big number, then [itex]x^{N}[/itex] will be very close to zero,right?

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- #40

arildno

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So, if we have [itex]S_{N}=\frac{1-x^{N}}{1-x}[/itex], |x|<1 and N is really big, what will [itex]S_{N}[/itex] be approximately equal to?

- #41

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Will Sn approximately be equal to = 1/1-x

Am I right?

Am I right?

- #42

arildno

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Thus, it gives perfect meaning to say that as N goes to infinity, Sn converges to a number S=1/(1-x), or that the INFINITE series S is a meaningful concept. Agreed?

- #43

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Ya I agree that now the S has a meaningful value.

- #44

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Oh please continue please? I can realy understand what you teach me than my school teacher.

- #45

arildno

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[tex]S=\sum_{n=0}^{\infty}x^{n}[/tex]

is a meaningful concept, as long as |x|<1.

We call 1 here to be the radius of convergence for the infinite series, that is the bound we must put on x, in order for the infinite series to have any meaning (I.e, being some number).

Okay?

- #46

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Anyway this is the first time I used this symbol next to the "Sn ="

Can you teach me about this also?

[tex]S=\sum_{n=0}^{\infty}x^{n}[/tex]

- #47

VietDao29

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[tex]\sum_{i = m} ^ n[/tex], this is the summation symbol. It's a capital Sigma.dilan said:

Anyway this is the first time I used this symbol next to the "Sn ="

Can you teach me about this also?

[tex]S=\sum_{n=0}^{\infty}x^{n}[/tex]

----------

[tex]\sum_{i = m} ^ n (a_i)[/tex]

means that all you need is just to sum from a

[tex]\sum_{i = m} ^ n (a_i) = a_m + a_{m + 1} + a_{m + 2} + ... + a_{n - 1} + a_n[/tex]

----------------

Example:

[tex]\sum_{k = 1} ^ 3 \left( \frac{k}{k + 1} \right) = \frac{1}{1 + 1} + \frac{2}{2 + 1} + \frac{3}{3 + 1} = \frac{23}{12}[/tex]

----------------

For

If you say some series [tex]S_N = \sum_{k = 1} ^ N (a_k)[/tex] converges to some number L, then it means that: [tex]\lim_{N \rightarrow \infty} S_N = L, \ N \in \mathbb{N}[/tex]

Can you get this? :)

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- #48

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arildno said:

[tex]S=\sum_{n=0}^{\infty}x^{n}[/tex]

is a meaningful concept, as long as |x|<1.

We call 1 here to be the radius of convergence for the infinite series, that is the bound we must put on x, in order for the infinite series to have any meaning (I.e, being some number).

Okay?

Hi VietDao29 thanks for your post about the meaning of the Capital letter of zigma with the other letters. Realy useful. Thanks alot.

Ya now I undertand. Okay we have now got a meaningful concept for Sn. Earlier I had a little problem in undertanding the symbol, but now I get it. Okay now we have a meaningful concept as long as |x|<1

But I don't know anything about this

[tex]\lim_{N \rightarrow \infty} S_N = L, \ N \in \mathbb{N}[/tex]

Anyway arildno I hope that this will not be needed for now. If yes I would like to know about that also.

Okay upto here now I can get it. What's the next step?

- #49

VietDao29

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Have you learnt limit by the way? Something looks like:dilan said:But I don't know anything about this

[tex]\lim_{N \rightarrow \infty} S_N = L, \ N \in \mathbb{N}[/tex]

[tex]\lim_{x \rightarrow 3} x ^ 2 = 9[/tex] (this is an example of limit of a function)

or:

[tex]\lim_{n \rightarrow \infty} \frac{1}{n} = 0[/tex] (an example of limit of a sequence)???

- #50

arildno

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[itex]N\in\mathbb{N}[/itex] just means that N is a natural number (1, 2, 350000)and so on (not a fraction or decimal number).

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