Exercise · ZW-COMPLEX-TAYLOR-SERIES-010 · ★★★ Advanced
The coefficient of \(z^3\) in the Taylor series of \(\dfrac{e^z}{1-z}\) at \(0\) is:
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- C. \(\tfrac83\)
Explanation
It is the product of series: \(a_3=1+1+\tfrac12+\tfrac16=\tfrac83\) (sum of the coefficients of \(e^z\) up to \(z^3\)).
Review the concept: Complex Taylor series
The Taylor series of a function \(f\) holomorphic on a disc \(|z-z_0|<R\) is the power series in \(z-z_0\), centred at \(z_0\), whose coefficients \(a_n\) come from the derivatives of \(f\) at \(z_0\).
\[f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n,\quad a_n=\frac{f^{(n)}(z_0)}{n!}\]
More practice problems: complex taylor series
- ZW-COMPLEX-TAYLOR-SERIES-001 The coefficient of z^3 in the Taylor series of e^z at z_0=0 is:
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- ZW-COMPLEX-TAYLOR-SERIES-003 The radius of convergence of the Taylor series of 11-z at 0 is:
- ZW-COMPLEX-TAYLOR-SERIES-004 What is the radius of convergence of the Taylor series of 1z^2+1 at z_0=0?
- ZW-COMPLEX-TAYLOR-SERIES-005 What if the centre is z_0=2? The radius for 1z^2+1 is:
- ZW-COMPLEX-TAYLOR-SERIES-006 The coefficient of (z-1)^3 in the Taylor series of 1/z at z_0=1 is:
- ZW-COMPLEX-TAYLOR-SERIES-007 The radius of convergence of the Taylor series of tan z at 0 is:
- ZW-COMPLEX-TAYLOR-SERIES-008 If f is entire, its Taylor series at 0 converges to f(z):