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7.(a) (4 points) Using the geometric series, obtain a power series with centerc= 0 for the function1x+ 1.(b) (4 points) Using the properties of derivatives and integrals of power series, find a power series withcenterc= 0 forln(1 +x2).Be sure to state the radius of convergence in both cases carefully, and to justify your choice ofintegration constant if necessary.Solution:
UCLA: Math 31BFinal exam (practice) (solutions)Spring, 20178.(a) (3 points) Does the series∞Xn=014n2-1converge? Justify your answer carefully.(b) (7 points) Evaluate the series above.Hint:use the definition of the series as a limit of partialsums. Calculate the partial sum by first using partial fractions.Solution: