Express $\frac{1}{z+1}$ as a power series about z=1

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SUMMARY

The discussion focuses on expressing the function $\frac{1}{z+1}$ as a power series centered at $z=1$. The user correctly identifies the Maclaurin series for $\frac{1}{z+1}$ but seeks to convert it into a Taylor series about $z=1$. The transformation involves rewriting the function as $\frac{1}{2 + (z - 1)}$ and applying the formula for the geometric series. This results in the series expansion that converges around the point $z=1$.

PREREQUISITES
  • Understanding of Taylor series and power series expansions
  • Familiarity with geometric series and their convergence
  • Basic knowledge of complex functions and their representations
  • Proficiency in algebraic manipulation of rational functions
NEXT STEPS
  • Study the derivation of Taylor series for various functions
  • Learn about the convergence criteria for power series
  • Explore the geometric series and its applications in function expansion
  • Investigate the differences between Maclaurin and Taylor series
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Mathematicians, students studying calculus, and anyone interested in series expansions of functions in complex analysis.

Poirot1
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express $\frac{1}{z+1}$ as a power series about z=1. My working: I know $\frac{1}{z+1}=1-z+z^2-z^3+...$ but this is macluarin series.
 
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Write $\frac 1{1+z}=\frac 1{2+(z-1)}=\frac 12\frac 1{1+\frac{z-1}2}$.
 

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