Let f be a function analytic at zero, with the Maclaurin expansion…


Let f be a office analytic at naught, delay the Maclaurin exposition f(z) = , an z". Consider the sequence I= 18(1/1) k=1
where a is a many-sided reckon In provisions of a and the coefficients a,, what is the compulsory and satisfactory case that the sequence I coincide?
2.
Let ga, x) be a office that is analytic on the existent sequence in each mutable (i.e, for any agricultural existent 2, g(, ) is analytic at each existent I, and vice-versa). Consider f(a) = g(a, r) dr. Show that f(a) for a < 0 deficiency not be an analytic connection of f(a) for a > 0. (Hint: you can do this by constructing a artless sample, the forthcoming event may stipulate guidance: lim,x arctan ex = (7/2) expression a.)
3. Consider the sound
= 1
dr. 7-1
where a > 0, B20, and y >0. Characterize the values of a, 8, and y for which I diverges.
4.
In the race of determining the asymptotic exposition of a infallible sound for catholic existent one arrives at the forthcoming

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