(a) Professor Willoughbg is doing some work with innite sums and has made the following deduction: 1 He knows that the MacLaurin series for 1 can be written as 1+m+m2 +:v3 +3:4 +- ' ' , so he I ll: multiplies both sides by :r to get a MacLaurin series for the function a: _ 1 = :I:a:2:c3- . - . The Professor uses the value ofx = 2 and gets the infinite sum, 2 4 8 - . - . The terms of this sum are tending to 00, so Professor Willoughby uses the nth Term Test to conclude that a: _ 1 , Willoughby deduces that the value of this innite sum must be f(2) = 2. Since the sum has a value, it must the sum must diverge. But, since he just used a: = 2 in the series for f(a:) = converge rather than diverge. Eaplain the error in Professor Willoughby's reasoning. Does the sum diverge or converge? What did he do wrong to think otherwise? (b) Consider the power series, 2.2,,(3; a)". Suppose we know that the radius of convergence of this series n=0 00 c is some nonzero real number, R. Determine whether the series 2 m converges absolutely, converges 11:0 Ra 2 conditionally, or diverges for a: = R . {c} Professor Willoughhy is still interested in the function so he has come up with another deduction: Il' 1 He starts again with the series for , which can be written as 1+:1:+:r2 +s:3 +324 + ' ' - . The 1 t 1 1 1 1 l : l I = 1 _ _ _ _ . . . ' Professor decides to let a: t and nds an innite sum, t _ 1 + t + t2 + 1531+ t41+ 1 To check his work, Professor Willoughby tests with a value oft = 2 and nds 2 = 1+5 +3 + g +- . ' . Professor Willoughby uses his knowledge of geometric series and is relieved that the equation at the end is correct. 1 1 1 1 Are his observations correct, and if so what other values oft makes the series 1 + t + t_2 + 133 + 734 + converge? Or has Professor Willoughby made another mistake, and if so explain the mistake in detail by referencing tests and theorems we learned in our lessons