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Converging series

AEA.PNG
Firstly what does converge mean exactly; I think I know what it means but I'm not 100% sure. I believe it means when the a series goes towards a number.
Any ways I don't understand how when -1 < x < 3 it converges. When x = 2 I think it goes to infinity as all the terms are being added up ( you get 1 + 0.5 + 0.25 +... numbers get smaller).
As the number of terms tends to infinite, the summation will tend towards one value if the series converges.

I'm afraid I can't help with the actual question - I don't know that much about sequences and series.
Reply 2
Avatar for ztibor
ztibor
10
Original post by WinOrDie
AEA.PNG
Firstly what does converge mean exactly; I think I know what it means but I'm not 100% sure. I believe it means when the a series goes towards a number.
Any ways I don't understand how when -1 < x < 3 it converges. When x = 2 I think it goes to infinity as all the terms are being added up ( you get 1 + 0.5 + 0.25 +... numbers get smaller).


Convergence means existing finite limit.

For function series this means a limit function.
For
∑an⋅fn(x−c)\sum a_n\cdot f_n(x-c)
the radius of convergence around c is

1ρ=liman+1an=lim2r2r+1=12\displaystyle \frac{1}{\rho}=lim \frac{a_{n+1}}{a_n}=lim \frac{2^r}{2^{r+1}}=\frac{1}{2}

So ρ=2\rho =2

So the series convergent around 1 with radius of 2, that is
∣x−1∣<2→−2<x−1<2→−1<x<3\displaystyle |x-1|<2 \rightarrow -2<x-1<2 \rightarrow -1<x<3
(edited 10 years ago)
Reply 3
Avatar for WinOrDie
WinOrDie
OP
1
Original post by ztibor
Convergence means existing finite limit.

For function series this means a limit function.
For
∑an⋅fn(x−c)\sum a_n\cdot f_n(x-c)
the radius of convergence around c is

1ρ=liman+1an=lim2r2r+1=12\displaystyle \frac{1}{\rho}=lim \frac{a_{n+1}}{a_n}=lim \frac{2^r}{2^{r+1}}=\frac{1}{2}

So ρ=2\rho =2

So the series convergent around 1 with radius of 2, that is
∣x−1∣<2→−2<x−1<2→−1<x<3\displaystyle |x-1|<2 \rightarrow -2<x-1<2 \rightarrow -1<x<3


what does [Sum off] An.Fn(x-c) mean?
(edited 10 years ago)
Reply 4
Avatar for ztibor
ztibor
10
Original post by WinOrDie
what does [Sum off] An.Fn(x-c) mean?


a function series is a series, where the summands are not just real or complex numbers but functions.
Examples of function series include power series, Laurent series, Fourier series, etc.

For power series
http://en.wikipedia.org/wiki/Power_series
Reply 5
Avatar for davros
davros
16
Original post by WinOrDie
AEA.PNG
Firstly what does converge mean exactly; I think I know what it means but I'm not 100% sure. I believe it means when the a series goes towards a number.
Any ways I don't understand how when -1 < x < 3 it converges. When x = 2 I think it goes to infinity as all the terms are being added up ( you get 1 + 0.5 + 0.25 +... numbers get smaller).


Are you studying this as part of a course or have you just found this question at random?

You're asking about series convergence but you don't seem terribly clear on what the question's asking, so I'm not sure what level you're working at :smile:

Does "geometric series" ring any bells?

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