Find the series radius and interval of convergence. Find the values of x for whi

Question

Find the series radius and interval of convergence. Find the values of x for which the series converges (b) absolutely and (c) conditionally. sigma^infinity_n=1 (4x-1)^2n+1/n^3/2 The radius of convergence is. (Simplify your answer.) Determine the interval of convergence. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The interval of convergence is. (Type a compound inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) The series converges only at x=. (Type an integer or a simplified fraction.) The series converges for all values of x. For what values of x does the series converge absolutely? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Explanation / Answer

an=(4x-1)2n +1/n3/2

an+1=(4x-1)2n +3/(n+1)3/2

ratio test for convergence:

limn->|an+1/an|<1

limn->|[(4x-1)2n +3/(n+1)3/2]/[(4x-1)2n +1/n3/2]|<1

limn->|[(4x-1)2/(1+(1/n))3/2]|<1

|[(4x-1)2/(1+0)3/2]|<1

|(4x-1)2|<1

|(4x-1)|<1

|(x-(1/4))|<1/4

radius of convergence =1/44

-1/4 <(x-(1/4))<1/4

0<x<1/2

at x=0 series is n=1 to (-1)2n +1/n3/2=n=1 to -1/n3/2 converges by alternating series test

at x=1/2 series is n=1 to 1/n3/2 converges by p series test , p=1/2

interval of convergence is [0,1/2]

b) faor all these values of x series converges absolutely

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