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 Exercise 4.5 (one of the simple ones) 
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Joined: 26 Mar 2008, 17:06
Posts: 2
Post Exercise 4.5 (one of the simple ones)
Can someone help me with exercise 4.5, which asks about an elementary reason for the simple relationship between the two following power series:
1) 1 + x^2 + x^4 + x^6 ...
2) 1 - x^2 + x^4 - x^6


26 Mar 2008, 17:10

Joined: 26 Mar 2008, 21:50
Posts: 1
Post Re: Exercise 4.5 (one of the simple ones)
yeah i couldnt figure that out too


26 Mar 2008, 21:54

Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India
Post Re: Exercise 4.5 (one of the simple ones)
P(x)=1+x^2+x^4+x^6+...=\frac{1}{1-x^2}
Q(x)=1-x^2+x^4-x^6+...=\frac{1}{1+x^2}=P(ix)
This would suffice.


01 Apr 2008, 06:44

Joined: 22 Jan 2009, 19:33
Posts: 2
Post Re: Exercise 4.5 (one of the simple ones)
Sameed’s solution is perfect, but it was no evident to me. So I developed it:

\\<br />{i}^{2} = -1 \\<br />{i}^{3} = {i}^{2} * i = -i \\<br />\\<br />{i}^{4} = {i}^{3} * i = +1 \\<br />{i}^{5} = {i}^{4} * i = +i \\<br />{i}^{6} = {i}^{5} * i = -1 \\<br />{i}^{7} = {i}^{6} * i = -i \\<br />\\<br />{i}^{8} = {i}^{7} * i = +1 \\

etc.

In general:

\\<br />{i}^{4n+0}= +1 \\<br />{i}^{4n+1}= +i \\<br />{i}^{4n+2}= -1 \\<br />{i}^{4n+3}= -i \\<br />{i}^{4n+4}= +1 \\

etc.

Now we have:

\\<br />P(ix)=1+{(ix)}^{2}+{(ix)}^{4}+{(ix)}^{6}+ ...\\<br />P(ix)=1-{x}^{2}+{x}^{4}-{x}^{6}+ ...

Perhaps this could be helpful for any other.

Manuel


27 Jan 2009, 18:53

Joined: 09 Jan 2010, 02:06
Posts: 1
Post Re: Exercise 4.5 (one of the simple ones)
Here's another one, not as easy as Sameed's but still elementary:

\frac{1}{1-x^{2}}=1+x^2+x^4+\cdots

Change x^2 \rightarrow x^4 to get

\frac{1}{1-x^4}=1+x^4+x^8+\cdots

Now multiply both sides by 1-x^2:

\frac{1-x^2}{1-x^4}=\frac{1}{1+x^2}=(1-x^2)(1+x^4+x^8+\cdots)

Multiplying the RHS of the equation:

(1-x^2)(1+x^4+x^8+\cdots)=1+x^4+x^8+\cdots-x^2-x^6-\cdots

Finally, rearranging:

\frac{1}{1+x^2}=1-x^2+x^4-x^6+\cdots

This solution doesn't use complex numbers, which are introduced later in the book


14 Feb 2010, 03:39
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