Answer the questions below without performing long series manipulations.

CHEMISTRY · Multi-part · ISC · Chemistry · Coordination Compounds · Electrochemistry

Problem

Answer the questions below without performing long series manipulations. (1) For the function

p(x)=\displaystyle \frac{1}{1+5x}

, explain why its Maclaurin series converges only for

|x|<\tfrac15

. In your explanation, reference the relationship between the magnitude of

x

and the terms of the series. (2) The function

r(x)=\displaystyle \frac{1}{1-2x}

can be expanded about

x=0

by an appropriate transformation of the well-known series for

\tfrac1{1-x}

. Describe this transformation verbally and state the radius of convergence of the resulting series, justifying your answer.