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Exercise 4.5 (one of the simple ones)

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Exercise 4.5 (one of the simple ones)
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puffin88 Joined: 26 Mar 2008, 17:06 Posts: 2	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

karzom Joined: 26 Mar 2008, 21:50 Posts: 1	Re: Exercise 4.5 (one of the simple ones) yeah i couldnt figure that out too
26 Mar 2008, 21:54

Sameed Zahoor Joined: 12 Mar 2008, 10:57 Posts: 69 Location: India	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

Cabedo Joined: 22 Jan 2009, 19:33 Posts: 2	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

AlvaroJ Joined: 09 Jan 2010, 02:06 Posts: 1	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 : $\frac{1-x^2}{1-x^4}=\frac{1}{1+x^2}=(1-x^2)(1+x^4+x^8+\cdots)$ 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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