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1. Telescoping sum of odd-numerator fractions ●●○○○ 20 ptsCalculate $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\frac{11}{900}+\frac{13}{1764}+\frac{15}{3136}$. Give your answer as a decimal.
2. Contour integral of 1/(z^2+1) ●●○○○ 20 ptsCompute the complex contour integral $\displaystyle\int_{|z|=2}\frac{1}{z^2+1}\,dz$, where the circle $|z|=2$ is traversed once counterclockwise.
3. Principal value with real poles ●●●○○ 50 ptsFind the Cauchy principal value of $\int_{-\infty}^{\infty}\frac{x\,dx}{(x-1)^4-1}$, where the poles at $0$ and $2$ lie on the real axis. Round to two decimal places.
4. Complex line integral of 1/(1+z^2) ●●●○○ 50 ptsEvaluate $\int_1^{1+i}\frac{1}{1+z^2}\,dz$ along the straight line segment from $z=1$ to $z=1+i$. Give the real part and the imaginary part of the result separately. Round each to two decimal places.