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Wendi Zhao is now a Ph.D. candidate in Department of Economics at Pennsylvania State University. Prior to this, Zhao acquired a Master's degree in Economics from Peking University and a Bachelor's degree in Life Science from Wuhan University, China. Zhao's research interests focus on the interdiscipline of economics and machine learning. Zhao tries to use methods from data science to deals with economic data and interpret phenomena in financial market. Except for courses in economics, Zhao has also taken many courses in math and computer science, such as real analysis, complex analysis, functional analysis, abstract algebra, reinforcement learning, probabilisitic graphical models, and algorithms.

A point inside the production possibilities frontier isa. efficient but not feasible.b. feasible but not efficient.c. both efficient and feasible.d. neither efficient nor feasible.

If $$f(x)=\Sigma_{n-0}^{\infty} b_{n}(x-5)^{n} \text { for all } x, \text { write a formula for } b_{8}$$

If $f^{(n)}(0)=(n+1) !$ for $n=0,1,2, \ldots,$ find the Maclaurin series for $f$ and its radius of convergence.

Use the definition of a Taylor series to find the first fournonzero terms of the series for $f(x)$ centered at the given value of $a$ .$$f(x)=(1-x)^{-2}$$

Find the Maclaurin series for $f(x)$ using the definition ofa Maclaurin series. [ Assume that $f$ has a power series expansion.Do not show that $R_{n}(x) \rightarrow 0 .$ J Also find the associated radius ofconvergence.$$f(x)=(1-x)^{-2}$$

Find the Maclaurin series for $f(x)$ using the definition ofa Maclaurin series. [ Assume that $f$ has a power series expansion.Do not show that $R_{n}(x) \rightarrow 0 .$ J Also find the associated radius ofconvergence.$$f(x)=\ln (1+x)$$