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Bernoulli numbers:Consider the functionf(z) =zez−1.(a) Show thatf(z) has a removable singularity atz= 0. Assume from now onthat the definition off(z) has been extended to remove the singularity.(b) Suppose you were to find a Taylor series forf(z), centered atz= 0. Whatwould be its radius of convergence?(c) Find the Taylor series in the formf(z) =X∞n=0Bnn!zn.The numbersBnare known as the Bernoulli numbers.(d) Find a recursion formula for the Bernoulli numbers, and use it to findB0, . . . ,B12.(e) Show thatB2n+1= 0 forn≥1.(f) Use your result to find a Taylor series forzcothz, in terms of the Bernoullinumbers. Where is this series valid? Using this result, find a Laurent seriesfor cotz. Where is this series valid?