How does one find the center of a power series?

What do you mean? Okay let's suppose we have the following:

  • sum(n=1 to infty) (x - 1)^2 / n^3
What is this centered at, and how does one find it? I need exact steps. Most math books do not explain things conceptually to understand. How do I go about finding the center of this using the following?

  • sum(n=0 to infty) a[sub:1imejfeh]n[/sub:1imejfeh]x[sup:1imejfeh]n[/sup:1imejfeh] = a[sub:1imejfeh]0[/sub:1imejfeh] + as[sub:1imejfeh]1[/sub:1imejfeh]x + a[sub:1imejfeh]2[/sub:1imejfeh]x[sup:1imejfeh]2[/sup:1imejfeh] + ... + a[sub:1imejfeh]n[/sub:1imejfeh]x[sup:1imejfeh]n[/sup:1imejfeh] + ...
or:
  • sum(n=0 to infty) a[sub:1imejfeh]n[/sub:1imejfeh](x - c)[sup:1imejfeh]n[/sup:1imejfeh] = a[sub:1imejfeh]0[/sub:1imejfeh] + as[sub:1imejfeh]1[/sub:1imejfeh](x - c) + a[sub:1imejfeh]2[/sub:1imejfeh](x - c)[sup:1imejfeh]2[/sup:1imejfeh] + ... + a[sub:1imejfeh]n[/sub:1imejfeh](x - c)[sup:1imejfeh]n[/sup:1imejfeh] + ...
I was thinking the one form I should use is the second one so is it centered at x = -2 or x = 2 in the book says positive and I do not get it why.

Consider this website http://www.ltcconline.net/greenl/courses/117/seqSerNewton/pow.htm, example 2: It has to be positive one if you say it is centered around a then it has to be 2, they got -2. How do they figure out -2? I just need all the steps how they have done it.
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Edited by stapel -- Reason for edit: formatting, etc.
 
CHAMI_NIRO said:
What do you mean? <<< I completely fail to understand your confusion. The referenced web-site was very explicit in its definition of center - and I thought so was I.

...suppose we have the following:

  • sum(n=1 to infty) (x - 1)^2 / n^3
What is this centered at, and how does one find it?

You have expansion with (x-2). If you compare this with (x-a) - what is a =??

a = +2

So it is centered around +2

How do I go about finding the center of this using the following?

  • sum(n=0 to infty) a[sub:sbh3fcnj]n[/sub:sbh3fcnj]x[sup:sbh3fcnj]n[/sup:sbh3fcnj] = a[sub:sbh3fcnj]0[/sub:sbh3fcnj] + as[sub:sbh3fcnj]1[/sub:sbh3fcnj]x + a[sub:sbh3fcnj]2[/sub:sbh3fcnj]x[sup:sbh3fcnj]2[/sup:sbh3fcnj] + ... + a[sub:sbh3fcnj]n[/sub:sbh3fcnj]x[sup:sbh3fcnj]n[/sup:sbh3fcnj] + ...
You have expansion with (x-0). If you compare this with (x-a) - what is a =??

a = 0

So it is centered around 0

or:
  • sum(n=0 to infty) a[sub:sbh3fcnj]n[/sub:sbh3fcnj](x - c)[sup:sbh3fcnj]n[/sup:sbh3fcnj] = a[sub:sbh3fcnj]0[/sub:sbh3fcnj] + as[sub:sbh3fcnj]1[/sub:sbh3fcnj](x - c) + a[sub:sbh3fcnj]2[/sub:sbh3fcnj](x - c)[sup:sbh3fcnj]2[/sup:sbh3fcnj] + ... + a[sub:sbh3fcnj]n[/sub:sbh3fcnj](x - c)[sup:sbh3fcnj]n[/sup:sbh3fcnj] + ...
You have expansion with (x-c). If you compare this with (x-a) - what is a =??

a = +c

So it is centered around +c

I was thinking the one form I should use is the second one so is it centered at x = -2 or x = 2 in the book says positive and I do not get it why.

Consider this website http://www.ltcconline.net/greenl/courses/117/seqSerNewton/pow.htm, example 2....

You have expansion with (x+2). If you compare this with (x-a) - what is a =??

a = -2

So it is centered around -2
Click to expand...
 
Just to understand myself, if we have the following:

  • sum(n=1 to infty) [ (-1)[sup:14s23on6]n[/sup:14s23on6] (1*3*...*(2n-1)) ] x[sup:14s23on6]n[/sup:14s23on6] / [ 2[sup:14s23on6]n[/sup:14s23on6] n! ]
This series is centered at an x^n so then is this centered around 0 then? So you mean without any calculation knowing the therom we can see where it is centered at?
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Edited by stapel -- Reason for edit: fixing formatting, etc
 
CHAMI_NIRO said:
Just to understand myself, if we have the following:

  • sum(n=1 to infty) [ (-1)[sup:eek:6mzhznx]n[/sup:eek:6mzhznx] (1*3*...*(2n-1)) ] x[sup:eek:6mzhznx]n[/sup:eek:6mzhznx] / [ 2[sup:eek:6mzhznx]n[/sup:eek:6mzhznx] n! ]
This series is centered at an x^n so then is this centered around 0 then? So you mean without any calculation knowing the therom we can see where it is centered at?
Click to expand...

What is the definition of the center of expansion of a power series? --- Look up in your text book....
 
It shows two definitions.

sum(n=0 to infty) anxn = a0 + as1x + a2x2 + ... + anxn + ...
or:

sum(n=0 to infty) an(x - c)n = a0 + as1(x - c) + a2(x - c)2 + ... + an(x - c)n + ...


With those two, with those two. Both of those are not the same. And do I have to match the form with the given form and then decide?
 
sum(n=1 to infty) [ (-1)n (1*3*...*(2n-1)) ] xn / [ 2n n! ]

Answer to the above question is that it is centerd at 0 right?
 
CHAMI_NIRO said:
It shows two definitions.

These are two examples. Definition was given in the line above (in the cited web-site)

sum(n=0 to infty) anxn = a0 + as1x + a2x2 + ... + anxn + ...
or:

sum(n=0 to infty) an(x - c)n = a0 + as1(x - c) + a2(x - c)2 + ... + an(x - c)n + ...


With those two, with those two. Both of those are not the same. And do I have to match the form with the given form and then decide? Yes
Click to expand...
 
CHAMI_NIRO said:
sum(n=1 to infty) [ (-1)n (1*3*...*(2n-1)) ] xn / [ 2n n! ]

Answer to the above question is that it is centerd at 0 <<< How did you come to that conclusion?

right?<<< Why do you doubt yourself?
Click to expand...
 
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