CVPL4

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Write the Taylor expansion of f(z)=z^{5}+3z+4\, at z=2\,.

In general, the Taylor expansion around z_{0}\, is f(z)=\sum _{{k=0}}^{\infty }{\frac  {f^{{(k)}}(z_{0})}{k!}}(z-z_{0})^{k}\,.

In this case, f'(z)=5z^{4}+3,f''(z)=20z^{3},f'''(z)=60z^{2},f^{{(4)}}(z)=120z,f^{{(5)}}(z)=120\,.

So

f(z)=42+83(z-2)+80(z-2)^{2}+40(z-2)^{3}+10(z-2)^{4}+(z-2)^{5}\,


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