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SOLVED:Find the convergence set for the given power series. ∑n=1^∞ n x^n

SOLVED:Find the convergence set for the given power series. ∑n=1^…

$SOLVED:Consider the series ∑n=0^∞ z^n, ∑n=1^∞ (t^n)/(n^2), and ∑n=1^∞ ( n^n)/(n). (a) Show that each series has radius of convergence 1 . (b) Show that the first series converges nowhere on C1(0)={z:|z|=1}. (c) Show that the second series converges everywhere on C1(0). (d) It turns out that the third series converges everywhere on C1(0), except at the point z=1. This is not easy to prove. Give it a try.$

SOLVED:Consider the series ∑n=0^∞ z^n, ∑n=1^∞ (t^n)/(n^2), a…

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SOLVED:Suppose that ∑n=1^∞ an is a convergent series of positive terms. Must the series ∑n=1^∞ √(an) also be convergent?

SOLVED:Suppose that ∑n=1^∞ an is a convergent series of positive ter…

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SOLVED:(a) Prove that the series ∑n=1^∞ ((-1)^n 1)/(2 n-1) converges. (b) Find the maximum error made in approximating the sum by the first 8 terms and the first 9 terms of the series. (c) How many terms of the series are needed in order to obtain an error which does not exceed .001 in absolute value? (a) The series is 1-(1)/(8) (1)/(6)-(1)/(4) (1)/(8)-⋯. If ux=((-1)^n 1)/(2 n-1), then an=|u*|=(1)/(2 n-1), an 1= |un 1|=(1)/(2 n 1). Since (1)/(2 n 1) ≦(1)/(2 n-1) and since limx →∞ (1)/(2 n-1)=0,

SOLVED:(a) Prove that the series ∑n=1^∞ ((-1)^n 1)/(2 n-1) converge…

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