Question

Consider the series (n=1 and infinite) ∑(−1)^(n+1) (x−3)^n / [(5^n)(n^p)], where p is a constant and p > 0.

a) For p=3 and x=8, does the series converge absolutely, converge conditionally, or diverge? Explain your reasoning.

b) For p=1 and x=8, does the series converge absolutely, converge conditionally, or diverge? Explain your reasoning.

c) When x=−2, for what values of p does the series converge? Explain your reasoning.

(d) When p=1 and x=3.1, the series converges to a value SS. Use the first two terms of the series to approximate S. Use the alternating series error bound to show that this approximation differs from S by less than (1/300,000).

Answer 1

Hence solved the question, if you have any dought please comment below and rate the question...thank you.

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ni м: си и Solution Civen, series a cight (x-3) - here x=-2 § (-) ht! (-5)" If the above series is convergent P31 otherwise it is divergent. only if Solution. Given, series, c-vjht! (3:1-3)* ouven, soves, hai son n' GP=1&x=3-1) an = wight" (o..sh as (ight! n. (5x10) ht+1 Antis AS It converges mer hapo (56) ht!(n+1) 1. Mal (sohn for n=1 a ti el a (-1) 41 213 cory. Sot! (141) 2500X2 ay -- sovo for n=2 de totes A2+1 = 4 = (-1)2+2 (56)2+(2+0) (33+) alternating = 3,75,000 1. Az = (ro)? 3 - 3,75,000 bounds aries (antil elanl 1 a, a, as are the first Age to the three terms of an - Az = zasado < 3,00,00D .: By alternating series bound, az < 3,00,000 DDR bor