C4 Binomial Expansion help!Watch

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Thread starter 6 years ago
#1
Holaaa,

Ok so I was just doing a bit of C4 today, and I dont understand how to get the range of a binomial expansion that doesnt converge, i.e. an expansion that has a power n which is -ve or a fraction (not a positive integer).

I understand that if i have (1+3x)^-2

then the range will be -1<3x<1

so that gives -1/3 < x < 1/3

which gives |x|<1/3 right?

HOWEVER.

the part I dont understand is why does the range of an expansion that doesn't converge start with -1<x<1 ? Why does the x value in a binomial expansion have to be between -1 and 1? I dont understand where that came from in the book

If someone could explain it to me I'd be very grateful

A
0
6 years ago
#2
Assuming that I've understood your question correctly, it's to do with the binomial series (1+x)^n (where n is not a positive integer) having a radius of convergence of 1. Which is something you don't need to worry about for C4 - it's something you'd come across in a first Analysis course at University.
1
Thread starter 6 years ago
#3
Thank you!
0
6 years ago
#4
(Original post by Aso)
Holaaa,

Ok so I was just doing a bit of C4 today, and I dont understand how to get the range of a binomial expansion that doesnt converge, i.e. an expansion that has a power n which is -ve or a fraction (not a positive integer).

I understand that if i have (1+3x)^-2

then the range will be -1<3x<1

so that gives -1/3 < x < 1/3

which gives |x|<1/3 right?

HOWEVER.

the part I dont understand is why does the range of an expansion that doesn't converge start with -1<x<1 ? Why does the x value in a binomial expansion have to be between -1 and 1? I dont understand where that came from in the book

If someone could explain it to me I'd be very grateful

A
The general binomial expansion is an infinite funtion series

where is a complex number.
When is an positive integer f.e n, then the (n+1)th and following terms are 0 in the series, so the sum is finite for any real x
For other alpha the series will be convergent or divergent depending on x value
f.e x=-1 the series converges if and only if
x>1 the series diverges ubless is non-negative integer
The question may be that for wich x will be the series convergent for any alpha
To answer this we have to calculate the radius of convergence of this function series around zero, because the terms
the radius can be calculated with the ratio

Using that

So
That is the series convergent for any alpha when -1<x<1
0
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