answer question number 5 please Consider the sequence given recursively by a1 =

Question

answer question number 5 please

Consider the sequence given recursively by a1 = 3 and an+1 = 1/4 - an. Suppose it is given that the sequence satisfies 0 an 3 for all n and is decreasing. Deduce the sequence is convergent and find the limit. How many terms of series 1/n(ln n)3 are needed to estimate the sum to 4 decimal places? Determine whether the following sequence is absolutely convergent, conditionally convergent or divergent. (-1)n/ln n Find a power series representation of f(x) = 1/1 + x2 and determine the interval of convergence.

Explanation / Answer

the next term (n+1)th to the nth term upto which all the terms are required can be at most 0.0009

so 1/(n+1)[ln(n+1)]^3 >= 0.00009

ie (n+1)[ln(n+1)]^3 <= 11111.11

By hit and trail we get n+1 = 108

thus n=107

so 107 terms of series are needed

(for hit and trail take numbers like 100 , 1000 and then narrow down your search)

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