Understanding an expansion into a geometric series

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SUMMARY

The discussion centers on the expansion of the function 1/(a-z) into a geometric series as described in Riley, Hobson, and Bence's "Mathematical Methods for Physics and Engineering." The specific expansion involves the variable substitution (z-z0)/(a-z0) and utilizes the Taylor series expansion around points other than zero. The introduction of z0 is crucial for understanding how to derive Taylor series at arbitrary points, enhancing the flexibility of series expansions in mathematical applications.

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  • Understanding of Taylor series and their applications
  • Familiarity with geometric series and their convergence
  • Basic knowledge of variable substitution in mathematical expressions
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  • Study the derivation of Taylor series around arbitrary points
  • Explore geometric series convergence criteria and applications
  • Review Riley, Hobson, and Bence's "Mathematical Methods for Physics and Engineering" for deeper insights
  • Practice variable substitution techniques in series expansions
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ThereIam
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Hi all,

I am reading through Riley, Hobson, and Bence's Mathematical Methods for Phyisics and Engineering, and on page 854 of my edition they describe (I am replacing variables for ease of typing)

"expanding 1/(a-z) in (z-z0)/(a-z0) as a geometric series 1/(a-z0)*Sum[((z-z0)/(a-z0))^n] for n = 0 to infinity."

I looked up the Taylor expansion of 1/(1-x) and see that it has a geometric form (Sum[x^n] also from 0 to infinity), but I do NOT see how or why they are introducing z0. Can someone explain explicitly what's going on here?

Much thanks!
 
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It looks like the authors wanted the students to learn how to get Taylor series around points other than 0.
 

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